A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

Abstract

This paper concerns the vertex reinforced jump process (VRJP), the edge reinforced random walk (ERRW), and their relation to a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential $\beta$ extending that defined by Sabot, Tarrès, and Zeng on finite graphs, and consider its associated random Schrödinger operator $H_\beta$. We construct a random function $\psi$ as a limit of martingales, such that $\psi =0$ when the VRJP is recurrent, and $\psi$ is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue $0$, when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function $\psi$, the Green function of the random Schrödinger operator, and an independent Gamma random variable. On ${\mathbb{Z}}^d$, we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension $d\ge 3$, using estimates of Disertori, Sabot, and Tarrès and of Disertori, Spencer, and Zimbauer. Finally, we deduce recurrence of the ERRW in dimension $d=2$ for any initial constant weights (using the estimates of Merkl and Rolles), thus giving a full answer to the question raised by Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator $H_\beta$.

1. Introduction

This paper concerns the Vertex Reinforced Jump Process (VRJP) and the Edge Reinforced Random Walk (ERRW) and their relation with a random Schrödinger operator associated with a stationary 1-dependent random potential (i.e., the potential is independent at distance larger than or equal to 2).

The VRJP is a continuous time self-interacting process introduced in 5, investigated on trees in 23 and on general graphs in 2021. We first recall its definition. Let ${\mathcal{G}}=(V,E)$ be an undirected graph with finite degree at each vertex. We write $i\sim j$ if $i\in V$, $j\in V,$ and $\{i,j\}$ is an edge of the graph. We always assume that the graph is connected and has no trivial loops (i.e., vertex $i$ such that $i\sim i$). Let $(W_{i,j})_{i\sim j}$ be a set of positive conductances, $W_{i,j}>0$, $W_{i,j}=W_{j,i}$. The VRJP is the continuous time process $(Y_s)_{s\ge 0}$ on $V$, starting at time $0$ at some vertex $i_0\in V$, which, conditionally on the past at time $s$, if $Y_s=i$, jumps to a neighbour $j$ of $i$ at rate

$$\begin{equation*} W_{i,j}L_j(s), \end{equation*}$$

where

$$\begin{equation*} L_j(s):=1+\int _0^s \mathds{1}_{\{Y_u=j\}}\,du. \end{equation*}$$

In 20 Sabot and Tarrès introduced the following time change of the VRJP

$$\begin{equation} Z_t=Y_{D^{-1}(t)}, {\tag{1.1}} \end{equation}$$

where $D(s)$ is the following increasing function

$$\begin{equation*} D(s)=\sum _{i\in V} (L^2_i(s)-1). \end{equation*}$$

We call this process the VRJP in exchangeable time scale and denote by ${{\mathbb{P}}}_{i_{0}}^{\text{VRJP}}$ its law starting from the vertex $i_0$. When the graph is finite, it is proved in Theorem 2 of 20 that the VRJP in exchangeable time scale $(Z_t)_{t\ge 0}$ is a mixture of Markov jump processes. More precisely, there exists a random field $(u_{j})_{j\in V}$ such that $Z$ is a mixture of Markov jump processes with jump rates from $i$ to $j$

$$\begin{equation*} \frac{1}{2}W_{i,j}e^{u_{j}-u_{i}}. \end{equation*}$$

The law of $(u_j)$ is explicit; cf. 20, Theorem 2 and the forthcoming Theorem B. It appears to be a marginal of a supersymmetric $\sigma$-field which had been investigated previously by Disertori, Spencer, and Zirnbauer (cf. 9, 10, 24). As a consequence of this representation and of 9, 10, it was proved in 20 that when the graph has bounded degree, there exists a real $\lambda _0>0$ such that if $W_{i,j}\le \lambda _0$ for all $i\sim j$, then the VRJP is positively recurrent; more precisely, $Z$ is a mixture of positive recurrent Markov jump processes. When the graph is the grid ${{\mathbb{Z}}}^d$, with $d\ge 3$, there exists $\lambda _1<+\infty$ such that if $W_{i,j}\ge \lambda _1$ for all $i\sim j$, the VRJP is transient. Hence, it shows a phase transition between recurrence and transience in dimension $d\ge 3$. The question of the representation of the VRJP on infinite graphs as a mixture of Markov jump processes is nontrivial, especially in the transient case. It is possible to prove such a representation by a weak convergence argument, following 16, but it gives little information on the mixing law. In this paper we prove such a representation involving the Green function and a generalized eigenfunction of a random Schrödinger operator.

Let us give a flavour of the main results of the paper in the case of the VRJP on ${{\mathbb{Z}}}^d$ with $W_{i,j}=W$ constant. We construct a positive 1-dependent random potential $(\beta _j)_{j\in {{\mathbb{Z}}}^d}$ (i.e., two subsets of the $\beta$’s are independent if their indices are at least at distance 2) and with marginal given by inverse of Inverse Gaussian law with parameters $({1\over dW},1)$. This field is a natural extension to infinite graphs of the field defined by Sabot, Tarrès, and Zeng in 22. We consider the random Schrödinger operator

$$\begin{equation*} H_\beta = -W\Delta +V, \end{equation*}$$

where $\Delta$ is the usual discrete (nonpositive) Laplacian and $V$ is the multiplication operator defined by $V_j= 2\beta _j -2dW$. Hence, it corresponds to the Anderson model with a random potential which is not i.i.d. but only stationary and 1-dependent. When the VRJP is transient, we prove that there exists a positive generalized eigenfunction $\psi$ of $H_\beta$ with eigenvalue 0, stationary and ergodic. Let $(G(i,j))_{i\in Z^d, j\in {{\mathbb{Z}}}^d}$ be defined by

$$\begin{equation*} G(i,j)= \widehat{G}(i,j)+\frac{1}{2}\gamma ^{-1} \psi (i)\psi (j), \end{equation*}$$

where $\widehat{G}=(H_\beta )^{-1}$ is the Green function (which happens to be well defined in an appropriate sense) and $\gamma$ is an extra random variable independent of the field $\beta$ with law $\Gamma (\frac{1}{2}, 1)$. We prove the following representation for the VRJP: the VRJP in exchangeable time scale $Z$ starting from the point $i_0$ is a mixture of Markov jump processes with jump rates from $i$ to $j,$

$$\begin{eqnarray} \frac{1}{2}W_{i,j} {G(i_0,j)\over G(i_0,i)}. {\tag{1.2}} \end{eqnarray}$$

When the VRJP is recurrent, the same representation is valid with $\psi =0$. In fact, the function $\psi$ is almost surely the limit of a martingale, the limit being positive when the VRJP is transient and 0 when the VRJP is recurrent. It is remarkable that when the VRJP is recurrent it can be represented as a mixture with $\beta$-measurable jump rates, but when the VRJP is transient it involves an extra independent Gamma random variable. This representation extends to infinite graphs the representation given in 22 for finite graphs. A new feature appears in the transient case, where the generalized eigenfunction $\psi$ is involved in the representation. We suspect that recurrence/transience of the VRJP is related to localization/delocalization of the random Schrödinger operator $H_\beta$ at the bottom of the spectrum.

The representation (1.2) has several consequences on the VRJP and the ERRW. The ERRW is a reinforced process introduced by Coppersmith and Diaconis in 1986 (see section 2.5 for a definition). The recurrence of the two-dimensional ERRW is a famous open question raised by Diaconis; see 4121718 for early references. Important progress has been made recently in the understanding of this process. In particular, in 20, an explicit relation between the ERRW and the VRJP was stated, thus somehow reducing the analysis of the ERRW to that of the VRJP. In 120, it was proved by rather different methods that the ERRW on any graph with bounded degree at strong enough reinforcement is positive recurrent. In 8, it was proved that the ERRW is transient on ${{\mathbb{Z}}}^d$, $d\ge 3$, at weak reinforcement.

The representation (1.2) allows us to complete the picture both in dimension 2 and in the transient regime. More precisely, we prove a functional central limit theorem for the ERRW and for the discrete time process associated with the VRJP in dimension $d\ge 3$ at weak reinforcement, using the estimates of 810. Using the polynomial estimate provided by Merkl and Rolles in 17, we are able to prove recurrence of ERRW on ${{\mathbb{Z}}}^2$ for all initial constant weights, hence giving a full answer to the question raised by Diaconis.

2. Statements of the results

2.1. Notation

We denote by ${{\mathbb{R}}}_+$ (resp. ${{\mathbb{R}}}^*_+$) the set of nonnegative (resp. positive) reals.

Let $\mathcal{G}=(V,E)$ be an undirected, locally finite, connected graph without trivial loops or multiple edges. For $i,j\in V$, write $i\sim j$ if $i$ is a neighbor of $j$. We write $\operatorname {d}_{\mathcal{G}}$ for the graph distance in $\mathcal{G}$, and for two subsets $U,U'$ of $V$, we define $\operatorname {d}_{\mathcal{G}}(U,U')=\inf _{i\in U,j\in U'}\operatorname {d}_{\mathcal{G}}(i,j)$. We suppose given, for each edge $e=\{i,j\}\in E$, a positive real $W_{i,j}>0$, which is understood as the conductance of the edge $e$. In this case we call $({\mathcal{G}}, (W_e)_{e\in E})$ a graph with conductances.

Convention.

We adopt the notation $\sum _{i\sim j}$ for the sum on all undirected edges $\{i,j\}$, counting each edge only once.

When $\beta =(\beta _i)_{i\in V}\in {{\mathbb{R}}}^V$ is a real vector indexed by the vertices and $U\subset V$, we write $\beta _U$ for the restriction of $\beta$ to $U$, i.e., $\beta _U=(\beta _i)_{i\in U}$. When $A=(A_{i,j})_{i,j\in V}\in {{\mathbb{R}}}^{V\times V}$ is a real function on $V\times V$ and $U\subset V$, $U'\subset V$, we write $A_{U,U'}$ for the restriction of $A$ to $U\times U'$, i.e., $A_{ U,U'}=(A_{i,j})_{i\in U,j\in U'}$.

It will be convenient to define the continuous time processes that appear in the text on the same canonical space. In the rest of this paper we will denote by ${D}([0,\infty ),V)$ the space of càdlàg functions from $[0,\infty )$ to $V$. The law of the VRJP in an exchangeable time scale, as defined in 1.1 and starting from $i_0$, will be denoted by ${{\mathbb{P}}}_{i_{0}}^{\text{VRJP}}$, which is a probability on ${D}([0,\infty ),V)$. The VRJP will always be defined on the canonical space, and $(Z_t)_{t\in {{\mathbb{R}}}_+}$ will denote the canonical process defined by $Z_t(\omega )=\omega (t)$ for $\omega \in {D}([0,\infty ),V)$.

Remark 1.

We do not allow multiple edges or trivial loops since it does not bring more generality to the VRJP. Indeed, from its definition, it follows that the VRJP on a graph with multiple edges and trivial loops has the same law as the VRJP on the graph where trivial loops are removed and multiple edges are replaced by a single edge by summing the conductances of the multiples edges. Similarly, the law on random potentials that appears in the rest of this paper can always be reduced to graphs without multiple edges or trivial loops. Nevertheless, in section 5 it simplifies notation to allow trivial loops.

2.2. Representation of the VRJP on infinite graphs

Define the operator $P=(P_{i,j})_{i,j\in V}$ by

$$\begin{equation*} P_{i,j}=\begin{cases} W_{i,j},& \text{if $i\sim j$}, \\ 0, & \text{otherwise}. \end{cases} \end{equation*}$$

We define below a probability distribution on potentials on the graph. A potential on the graph will generically be denoted $\beta =(\beta _i)_{i\in V}\in {{\mathbb{R}}}^V$. With the potential $\beta \in {{\mathbb{R}}}^V$, we associate the Schrödinger operator on $\mathcal{G}$

$$\begin{eqnarray} H_\beta = -P+2\beta , {\tag{2.1}} \end{eqnarray}$$

where $\beta$ represents the operator of multiplication by the potential $(\beta _i)$ (or equivalently the diagonal operator with diagonal terms $(\beta _i)_{i\in V}$).

We denote by

$$\begin{align} {\mathcal{D}}_V^W=\{\beta \in {{\mathbb{R}}}^V, \;\; (H_\beta )_{U,U}>0\text{ for all finite subsets $U\subset V$}\}, {\tag{2.2}} \end{align}$$

where $(H_\beta )_{U,U}>0$ means that the restriction of $H_\beta$ to $U\times U$ is positive definite. Obviously, ${\mathcal{D}}_V^W\subset ({{\mathbb{R}}}_+^*)^V$ since when $U=\{i\},$ the restriction of $H_\beta$ is the real $2\beta _i$. We endow ${\mathcal{D}}_V^W$ with its Borelian $\sigma$-field denoted ${{\mathcal{B}}}({\mathcal{D}}_V^W)$.

The following statement extends the random potential defined in 22, Theorem 1, to infinite graphs.

Proposition 1.

Let $({\mathcal{G}},(W_e)_{e\in E})$ be a graph with conductances as defined in section 2.1. There exists a unique probability distribution $\nu _{V}^W$ defined on $({\mathcal{D}}_V^W,{{\mathcal{B}}}({\mathcal{D}}_V^W))$, such that for any finite subset $U\subset V$ and any $(\lambda _i)_{i\in U}\in {{\mathbb{R}}}_{+}^U$:

$$\begin{equation*} \begin{split} &\int e^{-\sum _{i\in U} \lambda _i \beta _i} \nu _V^W(d\beta )\\ &\quad =e^{-\sum _{i\sim j, \; i,j\in U} W_{i,j}(\sqrt {(1+\lambda _i)(1+\lambda _j)}-1) -\sum _{i\sim j, i \in U, j\notin U} W_{i,j}(\sqrt {1+\lambda _i}-1)}{1\over \prod _{i\in U} \sqrt {1+\lambda _i}}. \end{split} \end{equation*}$$

In particular, on the probability space $({\mathcal{D}}_V^W,{{\mathcal{B}}}({\mathcal{D}}_V^W),\nu _V^W(d\beta ))$, we have the following properties:

•

1-dependence: If $U,U'\subset V$ are such that $\operatorname {d}_{\mathcal{G}}(U, U')\geq 2$, then the random variables $\beta \mapsto \beta _U$ and $\beta \mapsto \beta _{U'}$ are independent.

•

Reciprocal inverse Gaussian marginals: For $i\in V$, the random variable $\beta \mapsto \frac{1}{2\beta _i}$ has an inverse Gaussian distribution with parameter $(\frac{1}{W_{i}},1)$ where $W_i=\sum _{j\sim i} W_{i,j}$.

Remark 2.

On finite graphs, the density of $\nu _V^W$ is explicit; cf. 22, Theorem 1, and Theorem A below.

In the rest of this paper, the probability space $({\mathcal{D}}_V^W,{{\mathcal{B}}}({\mathcal{D}}_V^W),\nu _V^W)$ will be considered as the canonical space of random potentials on the graph. We write ${{\mathbb{E}}}_{\nu _V^W}$ for the expectation with respect to $\nu _V^W$. We will introduce several random variables on this probability space and adopt the following notation: when $\beta \mapsto X_\beta$ is a measurable function, we will write $X$ for the associated random variable and $X_\beta$ for its realization on the potential $\beta$. In particular, we will write $H$ for the random Schrödinger operator $\beta \mapsto H_\beta$ defined above. By abuse of notation, we sometimes consider $\beta _i$ for $i\in V$ or $\beta _U$ for $U\subset V$ as random variables (more precisely, the random variables are $\beta \mapsto \beta _i$ and $\beta \mapsto \beta _U$).

Definition 1.

Let $(V_{n})_{n\in {{\mathbb{N}}}}$ be an increasing sequence of finite connected subsets of $V$ such that

$$\begin{equation*} \bigcup _{n=0}^\infty V_n =V. \end{equation*}$$

For $n\in {{\mathbb{N}}}$, we define ${{\mathcal{F}}}^{(n)}\subset {{\mathcal{B}}}({\mathcal{D}}_V^W)$ as the sub-$\sigma$-field generated by the random variable $\beta \mapsto \beta _{V_n}$. For $n\in {{\mathbb{N}}}$ and $\beta \in {\mathcal{D}}_V^W$, we define a random operator $(\widehat{G}_\beta ^{(n)}(i,j))_{i,j\in V}$ by

$$\begin{equation*} \widehat{G}_\beta ^{(n)}(i,j) = \begin{cases} ((H_\beta )_{V_n,V_n})^{-1}(i,j) , & \text{ if $ i,j \in V_n$,} \\ 0, & \text{ otherwise.} \end{cases} \end{equation*}$$

For $n\in {{\mathbb{N}}}$ and $\beta \in {\mathcal{D}}_V^W$, we define a random function $(\psi _\beta ^{(n)}(i) )_{i\in V}$ as the unique solution of the equation

$$\begin{cases} H_\beta (\psi _\beta ^{(n)})(i)=0, &\text{ for $i\in V_n$,} \\ \psi _\beta ^{(n)}(i)=1, &\text{ for $i\in V^c_n$.} \end{cases}$$

By definition, the random variables $\widehat{G}^{(n)}:\beta \mapsto \widehat{G}^{(n)}_\beta$ and $\psi ^{(n)}:\beta \mapsto \psi ^{(n)}_\beta$ are ${{\mathcal{F}}}^{(n)}$-measurable.

The fact that there is a unique solution to the equation defining $\psi ^{(n)}_\beta$ is elementary; see the proof in section 4.2.

Our main theorem is the following.

Theorem 1.

(i)

For all $i,j\in V$, the sequence of random variables $\widehat{G}^{(n)}(i,j)$ is nondecreasing and converges a.s. to$$\begin{equation*} \widehat{G}(i,j):= \lim _{n\to \infty } \widehat{G}^{(n)}(i,j). \end{equation*}$$

Moreover, $\nu _V^W$-almost surely, $0<\widehat{G}(i,j)<\infty ,$ and the limit does not depend on the choice of the sequence of subsets $V_n$.

(ii)

Under the probability $\nu _V^W$, for all $i\in V$, $\psi ^{(n)}(i)$ is a positive ${{\mathcal{F}}}^{(n)}$-martingale. It converges a.s. to a random variable $\psi (i)$, such that $\psi (i)\ge 0$ a.s., and the limit does not depend on the choice of the increasing sequence $(V_n)$. Moreover, the quadratic variation of the vectorial martingale $(\psi ^{(n)}(i))_{i\in V}$ is given a.s. by$$\begin{equation*} \left<\psi (i),\psi (j)\right>_{n}=\widehat{G}^{(n)}(i,j). \end{equation*}$$

In particular, $\psi ^{(n)}(i)$ is bounded in $L^2$ if and only if ${{\mathbb{E}}}_{\nu _V^W}(\widehat{G}(i,i))<\infty$.

(iii)

For any real $\gamma >0$ and $\beta \in {\mathcal{D}}_V^W$, we define$$\begin{equation*} G_{\beta ,\gamma }(i,j)= \widehat{G}_\beta (i,j)+\frac{1}{2}\gamma ^{-1} \psi _\beta (i)\psi _\beta (j). \end{equation*}$$

For $i_0\in V$ and $x\in V$, denote by $P_x^{\beta ,\gamma ,i_{0}}$ the law of the Markov jump process which starts at $x\in V$ and jumps from $i$ to $j$ at rate$$\begin{eqnarray} \frac{1}{2}W_{i,j} \frac{G_{\beta ,\gamma } (i_{0},j)}{G_{\beta ,\gamma } (i_{0},i)}. {\tag{2.3}} \end{eqnarray}$$

Then the VRJP in an exchangeable time scale (defined in section 2.1 with conductances $(W_{i,j})$ and starting from $i_0)$ is a mixture of these Markov jump processes and has law$$\begin{equation} {{\mathbb{P}}}^{\text{VRJP}}_{i_{0}}(\ \cdot \ )=\int P_{i_0}^{\beta ,\gamma ,i_{0}}(\ \cdot \ ) \nu _V^{W}(d\beta ) \frac{\mathds{1}_{\gamma >0}}{\sqrt {\pi \gamma }}e^{-\gamma }d\gamma . {\tag{2.4}} \end{equation}$$

(iv)

For $\nu _V^W$-almost all $\beta$, all $\gamma >0,$ and all $i_0\in V$, we have

•

the Markov process $P_{i_0}^{\beta ,\gamma ,i_{0}}$ is transient if and only if $\psi _\beta (j)>0$ for all $j\in V;$

•

the Markov process $P_{i_0}^{\beta ,\gamma ,i_{0}}$ is recurrent if and only if $\psi _\beta (j)=0$ for all $j\in V$.

N.B.

Note that $P^{\beta ,\gamma ,i_0}_x$ is well defined for $\nu _V^W$-almost all $\beta$ and all $\gamma >0$ by (i) and (ii).

Notation.

We denote by

$$\begin{align} \nu _V^{W}(d\beta ,d\gamma ):=\nu _V^{W}(d\beta )\otimes \frac{\mathds{1}_{\gamma >0}}{\sqrt {\pi \gamma }}e^{-\gamma }d\gamma {\tag{2.5}} \end{align}$$

the probability distribution which appears in 2.4, under which $\gamma$ is $\Gamma (\frac{1}{2}, 1)$-distributed and independent of $\beta$. In general, we simply write $G(i,j)$ for $G_{\beta ,\gamma }(i,j)$ and consider it as a random variable on the probability space $({\mathcal{D}}_V^W\times {{\mathbb{R}}}^*_+, {{\mathcal{B}}}({\mathcal{D}}_V^W)\otimes {{\mathcal{B}}}({{\mathbb{R}}}_+^*), \nu _V^W(d\beta ,d\gamma ))$.

Remark 3.

When the VRJP is recurrent, then $G=\widehat{G}$, and the representation of the VRJP 2.4 only involves the variable $\beta$ and not $\gamma$.

Remark 4.

The representation (2.3) extends to infinite graphs the representation provided in 22, Theorem 2, for finite graphs. An interesting new feature appears in the transient regime, where the generalized eigenfunction $\psi$ and the extra gamma random variable enter the expression of $G(i,j)$. As it appears in the proof, the eigenfunction $\psi$ can be interpreted as the mixing field of a VRJP starting from infinity.

Denote by $\tau _{i_0}^{+}=\inf \{t\ge 0, \;\; Z_t=i_0, \; \exists s<t \text{ s.t. } Z_s\neq i_0\}$ the first return time to $i_0$ by $(Z_t)_{t\ge 0}$. The point Theorem 1(iv) is in fact a consequence of the following more precise assertion.

Proposition 2.

We have, for $\nu _V^W$-almost all $\beta$, for all $\gamma >0$ and $i_0$, $i\in V$,

$$\begin{equation*} P_{i}^{\beta ,\gamma ,i_{0}}(\tau _{i_0}^{+}=\infty )= \begin{cases} \frac{\psi (i_{0})^{2}}{4\gamma \widetilde{\beta }_{i_{0}}\widehat{G}(i_{0},i_{0})G(i_{0},i_{0})}, & \text{if $i=i_{0}$,}\\[6.0pt] \frac{\psi (i_{0})}{2\gamma }\frac{\widehat{G}(i_{0},i_{0})\psi (i)-\widehat{G}(i_{0},i)\psi (i_{0})}{\widehat{G}(i_{0},i_{0})G(i_{0},i)}, & \text{ if $i\neq i_{0}$,} \end{cases} \end{equation*}$$

where $\widetilde{\beta }_{i_{0}}=\sum _{j\sim i_{0}}\frac{1}{2}W_{i_{0},j}\frac{G(i_{0},j)}{G(i_{0},i_{0})}$. In particular, $\psi (i_0)=0$ if and only if

$$\begin{equation*} P_{i_0}^{\beta ,\gamma ,i_{0}}(\tau _{i_0}^{+}=\infty )=0. \end{equation*}$$

Using Doob’s $h$ transform, the law of the process $(Z_t)$ conditioned on the event $\{\tau _{0}^{+}<\infty \}$ or $\{\tau _{0}^{+}=\infty \}$ can be computed, and it takes a rather nice form, both under the law ${{\mathbb{P}}}^{\text{VRJP}}_{i_0}$ or under the law $P_{i_0}^{\beta ,\gamma ,i_{0}}$ for $\nu _V^W$-almost all $\beta$. We provide these formulae in section 7.

A natural question that emerges from point Theorem 1(iv) is that of a 0-1 law for transience/recurrence. We provide an answer below in the case of vertex transitive graphs with conductances. We say that $({\mathcal{G}}, W)$ is vertex transitive if the group of automorphisms of ${\mathcal{G}}$ that leaves invariant $(W_{i,j})$ is transitive on vertices. In particular, this is the case for the cubic lattice ${{\mathbb{Z}}}^d$ with constant conductances $W_{i,j}=W$. Denote by ${{\mathcal{A}}}$ the group of automorphisms that leave $W$ invariant.

Proposition 3.

If $({\mathcal{G}},W)$ is vertex transitive and ${\mathcal{G}}$ is infinite, then under the distribution $\nu _V^W(d\beta )$, the random variables $(\beta _i)_{i\in V}$, $(\psi (i))_{i\in V}$, $(\widehat{G}(i,j))_{i,j\in V}$ are stationary and ergodic for the group of transformations ${{\mathcal{A}}}$. Moreover, the VRJP is either recurrent or transient, i.e.,

$$\begin{equation*} {{\mathbb{P}}}^{\operatorname {VRJP}}_{i_0}( \text{ every vertex is visited infinitely often })=1 \end{equation*}$$

or

$$\begin{equation*} {{\mathbb{P}}}^{\operatorname {VRJP}}_{i_0}( \text{ every vertex is visited finitely often })=1. \end{equation*}$$

In the first case $\psi (i)=0$ for all $i\in V$, a.s., in the second case $\psi (i)>0$ for all $i\in V$, a.s.

N.B.

The action of ${{\mathcal{A}}}$ on $\widehat{G}$ is $(\tau \widehat{G})(i,j)= \widehat{G}(\tau i, \tau j)$ for $\tau \in {{\mathcal{A}}}$.

2.3. Relation with random Schrödinger operators

Let us now relate Theorem 1 to the properties of the random Schrödinger operator $H:\beta \mapsto H_\beta$ associated with the random potential $(\beta _j)$ under the law $\nu _V^W$, defined in 2.1 and Proposition 1.

Theorem 2.

Under $\nu _V^W(d\beta )$ the followoing hold:

(i)

The spectrum of $H$ is a.s. included in $[0,\infty )$.

(ii)

The operator $\widehat{G}$ is the inverse of $H$ in the following sense: for all $i,j\in V$ a.s.$$\begin{equation*} \widehat{G}(i,j)=\lim _{\epsilon >0, \epsilon \to 0} (H+\epsilon )^{-1}(i,j). \end{equation*}$$

(iii)

We have $(H \psi )(i)=0$ a.s. for all $i\in V$.

(iv)

In the case of the grid ${{\mathbb{Z}}}^d$ and when $W_{i,j}=W$ is constant, $(\widehat{G}(i,j))$ and $(\psi (i))$ are stationary ergodic for the spacial shift. Moreover, in the transient case, $\psi$ is a.s. a positive generalized eigenfunction with eigenvalue $0$ in the sense that $H \psi =0$ and $\psi$ has at most polynomial growth. More precisely, for all $p>d$ and $C>0$, a.s. there exists a random integer $K>0$ such that$$\begin{equation*} \vert \psi (i) \vert \le C \| i\|_\infty ^p \;\;\; \forall i\in {{\mathbb{Z}}}^d \text{ such that }\; \|i\|_\infty \ge K. \end{equation*}$$

2.4. Functional central limit theorem

We denote by $(\widetilde{Z}_{n})_{n\in {{\mathbb{N}}}}$ the discrete time process that describes the successive jumps of $(Z_{t})_{t\in {{\mathbb{R}}}_+}$. From Theorem 1(iii), under ${{\mathbb{P}}}^{\text{VRJP}}_{i_0}$, $\widetilde{Z}_{n}$ is a mixture of Markov chains starting from $i_0$ and with conductances

$$\begin{equation} W_{i,j}G(i_{0},i)G(i_{0},j) {\tag{2.6}} \end{equation}$$

under the probability distribution $\nu _V^W(d\beta ,d\gamma )$.

We prove below a functional central limit theorem for the discrete time VRJP on ${{\mathbb{Z}}}^d$, $d\ge 3$, at weak reinforcement (i.e., for $W$ large enough).

Theorem 3.

Consider the cubic graph $\mathbb{Z}^{d}$, $d\ge 3$, with constant conductances $W_{i,j}=W$. Denote

$$\begin{equation*} \widetilde{Z}^{(n)}_{t}=\frac{\widetilde{Z}_{[nt]}}{\sqrt {n}}. \end{equation*}$$

There exists $\lambda _2>0$ such that if $W>\lambda _2$, the discrete time VRJP satisfies a functional central limit theorem (i.e., under ${{\mathbb{P}}}_{0}^{\text{VRJP}})$ for any real $0<T<\infty$ and $(\widetilde{Z}^{(n)}_{t})_{t\in [0,T]}$ converges in law (for the Skorokhod topology) to a $d$-dimensional Brownian motion $(B_{t})_{t\in [0,T]}$ with nondegenerate isotropic diffusion matrix $\sigma ^2 \operatorname {Id}$, for some $0<\sigma ^2<\infty$.

2.5. Consequences for the ERRW

The ERRW is a famous discrete time process introduced in 1986 by Coppersmith and Diaconis 412.

Endow the edges of the graph ${\mathcal{G}}=(V,E)$ with some positive weights $(a_e)_{e\in E}$. Let $(X_n)_{n\in \mathbb{N}}$ be a random process that takes values in $V$, and let $\mathcal{F}_n=\sigma (X_0,\ldots ,X_n)$ be the filtration of its past. For any $e\in E$, $n\in \mathbb{N}$, let

$$\begin{equation} N_n(e)=a_e+ \sum _{k=1}^n\mathds{1}_{\{\{X_{k-1},X_k\}=e\}} {\tag{2.7}} \end{equation}$$

be the number of crossings of the (undirected) edge $e$ up to time $n$ plus the initial weight $a_e$.

Then $(X_n)_{n\in \mathbb{N}}$ is called the ERRW with starting point $i_0\in V$ and weights $(a_e)_{e\in E}$, if $X_0=i_0$ and, for all $n\in \mathbb{N}$,

$$\begin{equation} \mathbb{P}(X_{n+1}=j\nobreakspace |\nobreakspace \mathcal{F}_n)=\mathds{1}_{\{j\sim X_n\}}\frac{N_n(\{X_n,j\})}{\sum _{k\sim X_n} N_n(\{X_n,k\})}. {\tag{2.8}} \end{equation}$$

We denote by $\mathbb{P}^{\text{ERRW}}_{i_0}$ the law of the ERRW starting from the initial vertex $i_0$. We will assume that the ERRW is defined on the canonical space $V^{{\mathbb{N}}};$ i.e., that $(X_n)_{n\in {{\mathbb{N}}}}$ is the canonical process on $V^{{\mathbb{N}}}$.

Important progress has been made in the last ten years in the understanding of this process; cf., e.g., 181720. In particular, in was proved in 2012 by Sabot and Tarrès in 20 and Angel, Crawford, and Kozma in 1, on any graph with bounded degree at strong reinforcement (i.e., for $a_e<\widetilde{\lambda }_0$ for some fixed $\widetilde{\lambda }_0>0$), that the ERRW is a mixture of positive recurrent Markov chains. It was proved by Disertori, Sabot, and Tarrès in 8 that on ${{\mathbb{Z}}}^d$, $d\ge 3$, the ERRW is transient at weak reinforcement; i.e., for $a_e>\widetilde{\lambda }_1$ for some fixed $\widetilde{\lambda }_1<\infty$.

From Theorem 1 of 20, we know that the ERRW has the law of a VRJP in independent conductances. More precisely, consider $(W_e)_{e\in E}$ as independent random variables with gamma distribution with parameters $(a_e, 1)$. Consider the VRJP in conductances $(W_e)_{e\in E}$ and its underlying discrete time process $(\widetilde{Z}_n)$. Then the annealed law of $(\widetilde{Z}_n)$ (after expectation with respect to $W$) is that of the ERRW $(X_n)$ with initial weights $(a_e)$. Hence, we can apply Theorem 1 at fixed $W$ and then integrate on $W$. We thus consider the joint law $\widetilde{\nu }_V^{a}(dW,d\beta ,d\gamma )$ on $({{\mathbb{R}}}_+^*)^E\times ({{\mathbb{R}}}_+^*)^V\times {{\mathbb{R}}}_+^*$ obtained from $\nu _V^W(d\beta ,d\gamma )$ after randomization with respect to $W$. More formally, let $\widetilde{\nu }^{a}_V(dW)$ be the probability distribution on $({{\mathbb{R}}}_+^*)^E$ such that under $\widetilde{\nu }^a_V(dW)$ if the random variables $W\mapsto W_e$ are independent with gamma distribution with parameters $(a_e,1)$, then $\widetilde{\nu }_V^a(dW,d\beta ,d\gamma )$ is the probability distribution on $({{\mathbb{R}}}_+^*)^E\times ({{\mathbb{R}}}_+^*)^V\times {{\mathbb{R}}}_+^*$ such that for any bounded measurable test function $F$,

$$\begin{equation*} \int F(W,\beta ,\gamma ) \widetilde{\nu }_V^{a}(dW,d\beta ,d\gamma )=\int \left( \int F(W,\beta ,\gamma ) \nu _V^{W}(d\beta ,d\gamma )\right)\widetilde{\nu }_V^a(dW). \end{equation*}$$

In the rest of this paper, $\widetilde{\nu }_V^{a}(dW,d\beta )$, $\widetilde{\nu }_V^{a}(d\beta )$ will denote the corresponding marginal distributions, and $\widetilde{\nu }^a_V(dW)$ is the $W$ marginal. (By definition, $\widetilde{\nu }_V^{a}(dW,d\beta )$ is supported on the set of $(W,\beta )$ such that $\beta \in {\mathcal{D}}_V^W$.) From Theorem 1 we see that the ERRW starting from $i_0$ is a mixture of reversible Markov chains with conductances

$$\begin{eqnarray} x_{i,j}=W_{i,j}G(i_0,i)G(i_0,j), {\tag{2.9}} \end{eqnarray}$$

where $G$ is defined in Theorem 1 and $(W,\beta ,\gamma )$ are distributed according to $\widetilde{\nu }_V^{a}(dW,d\beta ,d\gamma )$. More formally, if $\widetilde{P}^x_{i_0}$ denotes the law of the Markov chain starting at $i_0$ and with conductances $(x_{i,j})_{i\sim j}$, then

$${{\mathbb{P}}}^{\text{ERRW}}_{i_0}(\cdot )=\int \widetilde{P}^{x}_{i_0}(\cdot )\widetilde{\nu }_V^a(dW,d\beta ,d\gamma ).$$

An important point is that we keep the 1-dependence of the field $\beta$, after taking expectation with respect to $W$.

Proposition 4.

Under $\widetilde{\nu }_V^{a}(d\beta )$, $(\beta _j)_{j\in V}$ is $1$-dependent: if $U, U' \subset V$ are such that $\operatorname {d}_{\mathcal{G}}(U, U')\geq 2$, then $(\beta _i)_{i\in U}$ and $(\beta _j)_{j\in U'}$ are independent.

Proof.

Indeed, from Proposition 1, the Laplace transform of $(\beta _i)_{i\in U}$ under $\nu _V^W(d\beta )$ only involves the conductances $W_{i,j}$ for $i$ or $j$ in $U$. This implies that, if $\operatorname {d}_{\mathcal{G}}(U, U')\geq 2$, the joint Laplace transform of $(\beta _i)_{i\in U}$ and $(\beta _i)_{i\in U'}$ is still the product of Laplace transforms even after taking expectation with respect to the random variables $(W_e)$, i.e., under $\widetilde{\nu }_V^a(\, \mathrm{d}\beta )$.

■

This yields a counterpart of Proposition 3 for the ERRW.

Proposition 5.

Assume $({\mathcal{G}},(a_{i,j}))$ is vertex transitive with automorphism group ${{\mathcal{A}}}$, and ${\mathcal{G}}$ infinite. Then under the distribution $\widetilde{\nu }_V^{a}(dW,d\beta )$, the random variables $(W_e)_{e\in E}$, $(\beta _i)_{i\in V}$, $(\psi (i))_{i\in V}$, $(\widehat{G}(i,j))_{i,j\in V}$ are stationary and ergodic for the group of transformations ${{\mathcal{A}}}$. Moreover, the ERRW is either recurrent or transient, i.e.,

$$\begin{equation*} {{\mathbb{P}}}^{\text{ERRW}}_{i_0}( \text{ every vertex is visited infinitely often })=1 \end{equation*}$$

or

$$\begin{equation*} {{\mathbb{P}}}^{\text{ERRW}}_{i_0}( \text{ every vertex is visited finitely often })=1. \end{equation*}$$

In the first case $\psi (i)=0$ for all $i\in V$ a.s. in the second case $\psi (i)>0$ for all $i\in V$ a.s.

N.B.

The action of ${{\mathcal{A}}}$ on $\widehat{G}$ and $W$ is $(\tau \widehat{G})(i,j)= \widehat{G}(\tau i, \tau j)$, $\tau W_{i,j}=W_{\tau i, \tau j}$ for $\tau \in {{\mathcal{A}}}$.

Remark 5.

In 16 it was proved on infinite graphs that the ERRW is a mixture of Markov chains, obtained as a weak limit of the mixing law of the ERRW on finite approximating graphs. The difference in the representation we give in (2.9) is that the random variables $\psi$, $\widehat{G}$ are obtained as almost sure limits and hence are measurable functions of the random variables $\beta$. This yields stationarity and ergodicity, which are the key ingredients in the 0-1 law, and in the forthcoming Theorems 4 and 5.

Remark 6.

It seems that this 0-1 law is new, both for the VRJP and the ERRW. In 16 it was proved that if the ERRW comes back with probability 1 to its starting point, then it visits infinitely often all points a.s., which is a weaker result. This was proved using the representation of the ERRW as mixture of Markov chains of 16. (A short proof of this last result can also be given, cf. 23.)

We now give a counterpart of Theorem 3 for the ERRW. It is a consequence of Theorem 1 and of the delocalization result proved by Disertori, Sabot, and Tarrès in 8.

Theorem 4.

Consider the cubic graph $\mathbb{Z}^{d}$, $d\ge 3$, with constant weights $a_{i,j}=a$. Denote

$$\begin{equation*} {X}^{(n)}_{t}=\frac{{X}_{[nt]}}{\sqrt {n}}. \end{equation*}$$

There exists $\widetilde{\lambda }_2>0$ such that if $a>\widetilde{\lambda }_2$, the ERRW satisfies a functional central limit theorem (i.e., under ${{\mathbb{P}}}_{0}^{\text{ERRW}}$ for any real $0<T<\infty )$ and $({X}^{(n)}_{t})_{t\in [0,T]}$ converges in law (for the Skorokhod topology) to a $d$-dimensional Brownian motion $(B_{t})_{t\in [0,T]}$ with nondegenerate isotropic diffusion matrix $\sigma ^2 \operatorname {Id}$, for some $0<\sigma ^2<\infty$.

Finally, we can deduce recurrence of the ERRW in dimension 2 from Theorem 1, Proposition 5, and the estimates obtained by Merkl and Rolles in 1517.⁠¹

¹

We are grateful to Franz Merkl and Silke Rolles for a useful discussion on that subject.

✖

Theorem 5.

The ERRW $({X}_{n})_{n\ge 0}$ on $\mathbb{Z}^{2}$ with constant weights $a_{i,j}=a$ is a.s. recurrent, i.e.,

$$\begin{equation*} {{\mathbb{P}}}^{\text{ERRW}}_{0}\left( \text{ every vertex is visited infinitely often }\right)=1. \end{equation*}$$

In 1517, by a Mermin–Wagner-type argument, Merkl and Rolles proved a polynomial decrease of the form

$$\begin{eqnarray} {{\mathbb{E}}}\left( \left({x_\ell \over x_0}\right)^{{1\over 4}} \right)\le c(a) \vert \ell \vert ^{-\xi (a)}, {\tag{2.10}} \end{eqnarray}$$

for some constants $c(a)>0$, $\xi (a)>0$, depending only on $a$, and where $x_{\ell }$ is the conductance at the site $\ell$ for the mixing measure of the ERRW, uniformly for a sequence of finite approximating graphs. When $0<\xi <1$, it does not give by itself enough information to prove recurrence. It was used in the case of a diluted two-dimensional graph to prove positive recurrence at strong reinforcement. The extra information given by the representation (2.9) and the stationarity of $\psi$ implies that the polynomial estimate (2.10) is incompatible with $\psi (i)>0$ and hence is incompatible with transience. Detailed arguments are provided in section 8.

Remark 7.

We expect similarly that the two-dimensional VRJP with constant conductances $W_{i,j}=W>0$ is recurrent. This would be implied by an estimate of the type (2.10) for the mixing field of the VRJP, which is still not available. More precisely, we can see from the proof of Theorem 5 in section 8 that recurrence of the two-dimensional VRJP would be implied by Theorem 1, Proposition 3, and an estimate of the type

$$\begin{equation*} {{\mathbb{E}}}\left(e^{\eta (u_\ell -u_0)}\right)\le \epsilon (\| \ell \|_{\infty }), \end{equation*}$$

for $\eta >0$ and $\epsilon (n)$ a positive function such that $\lim _{n\to \infty } \epsilon (n)=0$, where $(u_j)$ is the mixing field of the VRJP starting from 0 (cf. Theorem B) on finite boxes with wired boundary condition as in section 4.2. We learned from Kozma and Peled that they have a proof of such an estimate.

2.6. Open questions

The most important question certainly concerns the relation between the properties of the VRJP and the spectral properties of the random Schrödinger operator $H_\beta$. For example on ${{\mathbb{Z}}}^d$ with constant weights $W_{i,j}=W$, is recurrence/transience of the VRJP related to the localized/delocalized regimes of $H_\beta$? A more precise question would be, Does the transient regime of the VRJP coincide with the existence of extended states at least at the bottom of the spectrum of $H_\beta$? It might at first seem inconsistent to expect extended states at the bottom of the spectrum since the Anderson model with i.i.d. potential is expected to be localized at the edges of the spectrum (a fact which is proved in several cases). But this localization is a consequence of Lifshitz tails, and there are good reasons to expect that Lifshitz tails fail for the potential $\beta$, which is not i.i.d. but 1-dependent. Indeed, the bottom of the spectrum of $H_\beta$ is 0, and it does not coincide with the minimum of the support of the distribution of $2\beta$ translated by the spectrum of $-P$, as it is the case for i.i.d. potential. In fact, on a finite set, the minimum of the spectrum is reached on the set $\det (2\beta -P)=0$ which is a set of codimension 1, hence it is “big”.

Another natural question concerns the uniform integrability of the martingale $\psi ^{(n)}(i)$. Let us ask a more precise question: Is it true (at least for ${{\mathbb{Z}}}^d$ with constant weights) that transience of the VRJP implies that the martingale $\psi ^{(n)}(i)$ is bounded in $L^2$? It is quite natural to expect such a property from relation (5.2) since $\widehat{G}^{(n)}(i,i)$ appears to be the quadratic variation of $\psi ^{(n)}(i)$. This would have several consequences. First, it would imply that in dimension $d\ge 3$, the VRJP satisfies a functional central limit theorem as soon as the VRJP is transient, by the same argument as that of the proof of Theorem 3. It would also imply directly that the VRJP is recurrent as soon as the reversible Markov chain in conductances $(W_{i,j})$ is recurrent, if the group of automorphisms of $({\mathcal{G}},W)$ is transitive. Indeed, assume that the property is true and the VRJP is transient. By Theorem 1, the discrete time process $(\widetilde{Z}_n)$ would be represented as a mixture of reversible Markov chains with conductances $W_{i,j} G(0,i)G(0,j)$. From Proposition 2 applied to $i_0=0$, we have that

$$\begin{equation*} {\widehat{G}(0,i)\over \widehat{G}(0,0)} \le {\psi (i)\over \psi (0)}. \end{equation*}$$

Hence, $(\widetilde{Z}_n)$ is equivalently a mixture of Markov chains with conductances

$$\begin{equation*} \frac{\psi (0)^2}{G(0,0)^2} W_{i,j} {G(0,i)G(0,j)}\le W_{i,j} {\psi (i)\psi (j)}. \end{equation*}$$

But $(\psi (i))$ is stationary ergodic, if $\psi _0$ is squared integrable, we would have

$$\begin{equation*} {{\mathbb{E}}}_{\nu _V^W}(W_{i,j} {\psi (i)\psi (j)})\le CW_{i,j} \end{equation*}$$

for some constant $C>0$. Usual arguments imply that the Markov chain in conductance $W_{i,j}\psi (i)\psi (j)$ is recurrent if the Markov chain in conductances $(W_{i,j})$ is recurrent (cf., e.g., 14, Exercise 2.75). We arrive at a contradiction.

2.7. Organization of the paper

In section 3 we gather several results in the case of finite graphs, in particular we recall the main results of 22. In section 4 we define the important notion of restriction with wired boundary condition and the compatibility property. Section 5 is the key step in the paper where the martingale property is proved. In section 6 we prove Theorem 1, Propositions 2 and 3 and Theorem 2. In section 7 we provide extra computations of $h$-transforms. In section 8 we prove recurrence of ERRW in dimension 2 for all initial constant weights. In section 9 we prove functional central limit theorems for the VRJP and the ERRW, Theorems 3 and 4.

3. The random potential $\beta$ on finite graphs

In this section we assume that $\mathcal{G}=(V,E)$ is a finite graph and gather several results in this case. Recall that every undirected edge $e=\{i,j\}$ is labeled with a positive conductance $W_e=W_{i,j}$. In the case of a finite graph, the Schrödinger operator $H_\beta$ defined in 2.1 can be represented by the $V\times V$-matrix given by

$$\begin{equation*} H_{\beta }(i,j)= \begin{cases} 2\beta _i, & i=j,\\ -W_{i,j}, & i\neq j, \; i\sim j,\\ 0,&\text{otherwise,} \end{cases} \end{equation*}$$

and the set ${\mathcal{D}}_V^W$ defined in 2.2 is the set of potentials $\beta$ such that $H_\beta$ is positive definite.

3.1. The probability distribution $\nu _V^W$ on finite graphs, and its relation to the VRJP

We recall Theorem 1 from 22, which defines the probability distribution $\nu _V^W(d\beta )$ by its density on any finite graph.

Theorem A (22, Theorem 1, Definition 1, Proposition 1).

Let $(\mathcal{G},(W_e)_{e\in E})$ be a finite graph with conductances. The measure below is a probability on ${\mathcal{D}}_V^W$:

$$\begin{equation} \nu _V^{W}(d\beta ):=\mathds{1}_{H_{\beta }>0}\left(\frac{2}{\pi }\right)^{|V|/2}\exp ({-\sum _{i\in V}\beta _i+\sum _{e\in E}W_e})\frac{d\beta _{V}}{\sqrt {\det H_{\beta }}} {\tag{3.1}} \end{equation}$$

with $d\beta _{V}=\prod _{i\in V}d\beta _{i}$, and where $H_\beta >0$ means that $H_\beta$ is positive definite.

The Laplace transform of the probability distribution $\nu _V^W(d\beta )$ is given, for all $(\lambda _i)\in {{\mathbb{R}}}_+^V$, by

$$\begin{equation} \int e^{-\left<\lambda ,\beta \right>}\nu _V^{W}(d\beta )=\exp \left(-\sum _{i\sim j}W_{i,j}(\sqrt {(\lambda _{i}+1)(\lambda _{j}+1)}-1)\right) \prod _{i\in V}\frac{1}{\sqrt {\lambda _{i}+1} }. {\tag{3.2}} \end{equation}$$

Moreover, under $\nu _V^W(d\beta )$ we have the following properties:

•

1-dependence: If $U,U'\subset V$ are such that $\operatorname {d}_{\mathcal{G}}(U, U')\geq 2$, then the random variables $\beta \mapsto \beta _U$ and $\beta \mapsto \beta _{U'}$ are independent.

•

Reciprocal inverse Gaussian marginals: For $i\in V$, the random variable $\beta \mapsto \frac{1}{2\beta _i}$ has an inverse Gaussian distribution with parameter $(\frac{1}{W_{i}},1)$ where $W_i=\sum _{j\sim i} W_{i,j}$.

If we apply formula 3.2 to $(\lambda _i)\in {{\mathbb{R}}}_+^V$ such that $\lambda _{V\setminus U}=0$ for a subset $U\subset V$, we find the expression of Proposition 1. Hence, it implies Proposition 1 in the case of a finite graph.

The field $\beta$ is closely related to the VRJP, as shown in the next two theorems. In 20 it is shown that the VRJP in an exchangeable time scale defined in section 2.1 is a mixture of Markov jump processes; more precisely,

Theorem B (20, Theorem 2).

Assume $V$ finite. The following measure is a probability distribution on the set $\{(u_i)_{i\in V}\in \mathbb{R}^V,\; u_{i_0}=0\}$,

$$\begin{equation} \begin{split} \mathcal{Q}_{i_{0}}^{W}(du)&=\frac{1}{\sqrt {2\pi }^{|V|-1}}\\ &\quad \times \exp \left(-\sum _{i\in V}u_i-\sum _{i\sim j}W_{i,j}(\cosh (u_i-u_j)-1)\right)\sqrt {D(W,u)}du_{V\setminus \{i_{0}\}}, \end{split}{\tag{3.3}} \end{equation}$$

where $du_{V\setminus \{i_{0}\}}=\prod _{i\in V\setminus \{i_0\}} du_i$ and $D(W,u)=\sum _{T\in {\mathcal{T}}} \prod _{\{i,j\}\in T} W_{i,j}e^{u_i+u_j}$, where the sum is over ${\mathcal{T}}$, the set of spanning trees of the graph ${\mathcal{G}}$.

For $(u_i)_{i\in V}\in {{\mathbb{R}}}^V$, we denote by $P^{(u)}_{i_0}$ the law of the Markov jump process starting at vertex $i_0$ and with jump rates from $i$ to $j$ given by

$$\begin{equation*} \frac{1}{2}W_{i,j}e^{u_j-u_i}. \end{equation*}$$

The law of the VRJP in an exchangeable time scale starting at $i_0$ is a mixture of Markov jump processes, with mixing law given by

$${{\mathbb{P}}}^{\text{VRJP}}_{i_0}(\cdot )=\int P^{(u)}_{i_0}(\cdot ) \mathcal{Q}_{i_{0}}^{W}(du).$$

Remark 8.

By the matrix-tree theorem, $D(W,u)$ is any diagonal minor of the $|V|\times |V|$ matrix $(m_{i,j})$ with coefficients

$$\begin{equation*} m_{i,j}= \begin{cases} 0,& \text{ if } i\not \sim j,\; i\neq j,\\ - W_{i,j}e^{u_{i}+u_{j}}, & \text{ if } i\sim j,\; i\neq j,\\ \sum _{k\in V, k\sim i}W_{i,k}e^{u_{i}+u_{k}}, & \text{ if }i=j. \end{cases} \end{equation*}$$

Remark 9.

The probability measure $\mathcal{Q}_{i_{0}}^{W}(du)$ appeared previously to 20 in a rather different context in the work of Disertori, Spencer, and Zirnbauer 10. In particular, the fact that $\mathcal{Q}_{i_{0}}^{W}(du)$ is a probability measure was proved there as a consequence of a Berezin identity applied to a supersymmetric extension of that measure.

On finite graphs, the random environment $(u_i)$ of the previous theorem can be represented thanks to the Green function of the random potential $(\beta _{i},i\in V)$ distributed according to $\nu _V^W(d\beta )$. Let us first recall Proposition 1 of 22.

Proposition A (22, Proposition 1).

Assume $V$ finite. For $\beta \in {\mathcal{D}}_V^W$, we denote by

$$\begin{equation*} G_\beta :=(H_\beta )^{-1} \end{equation*}$$

the Green function of the Schrödinger operator $H_\beta$. For $\beta \in {\mathcal{D}}_V^W$, $i,j\in V$, we define $u_\beta (i,j)$ by

$$\begin{equation} e^{u_\beta (i,j)}=\frac{G_\beta (i,j)}{G_\beta (i,i)}. {\tag{3.4}} \end{equation}$$

For $i_0\in V$, $(u_\beta (i_0,j))_{j\in V}$ is the unique solution of the equation

$$\begin{align} \begin{cases} u_\beta (i_0,i_0)=0,& \\ \sum _{j\sim i} W_{i,j}e^{u_\beta (i_0,j)-u_\beta (i_0,i)}= \beta _i, &\text{if $i\neq i_0$}. \end{cases} {\tag{3.5}} \end{align}$$

In particular, the function $\beta \mapsto (u_\beta (i_0,j))_{j\in V}$ is $(\beta _j)_{j\in V\setminus \{i_0\}}$-measurable. Moreover, for all $\beta \in {\mathcal{D}}_V^W$,

$$\begin{equation} \beta _{i_0}=\frac{1}{2G_\beta (i_{0},i_{0})}+\frac{1}{2}\sum _{j\sim i_0}W_{i,j}e^{u_\beta (i_0,j)-u_\beta (i_0,i_0)}. {\tag{3.6}} \end{equation}$$

As usual, we simply denote by $G(i,j)$ and $u(i,j)$ the associated random variables on the probability space $({\mathcal{D}}_V^W,{{\mathcal{B}}}({\mathcal{D}}_V^W),\nu _V^W)$. Let us now recall 22, Theorem 3.

Theorem C (22, Theorem 3).

Assume $V$ finite. For all $i_0\in V$, under the probability $\nu _V^W(d\beta )$,

(i)

the random field $(u(i_0,j))_{j\in V}$ has the distribution $\mathcal{Q}_{i_{0}}^{W}$ of Theorem B;

(ii)

${1\over 2 G(i_0,i_0)}$ has a gamma distribution with parameters $(1/2, 1)$;

(iii)

$G(i_0,i_0)$ is independent of $(\beta _j)_{j\neq i_0}$, hence it is independent of the field $(u(i_0,j))_{j\in V}$.

Remark 10.

Here we only consider the VRJP with initial local time $1$, in fact, the above correspondence between $\beta$ and VRJP still holds for the process starting with any positive local times $(\phi _{i},i\in V)$, in such a case, there is a corresponding density $\nu _V^{W,\phi ^2}$, which is defined in 22, Definition 1 and Theorem 3. We choose here to normalize the initial local time to 1 since it is equivalent to the general case by a change of time and $W$; see 22, Appendix B.

Combining Theorem B and Theorem C gives a representation of the VRJP in an exchangeable time scale starting from different points in terms of the probability on random potentials $\nu _V^W$. We state this representation below.

Corollary 1.

Assume $V$ finite. For $\beta \in {\mathcal{D}}_V^W$, define $P^{\beta ,i_0}_x$ as the law of the Markov jump process starting from $x$ and with jump rates from $i$ to $j$ given by

$$\frac{1}{2}W_{i,j}{G_\beta (i_0,j)\over G_\beta (i_0,i)}.$$

Then the VRJP in exchangeable time scale is a mixture of the Markov jump processes

$$\begin{eqnarray} {{\mathbb{P}}}^{\text{VRJP}}_{i_{0}}(\ \cdot \ )=\int P_{i_0}^{\beta ,i_{0}}(\ \cdot \ ) \nu _V^{W}(d\beta ). {\tag{3.7}} \end{eqnarray}$$

3.2. Representation as a sum on paths

We call path in $\mathcal{G}$ from $i$ to $j$ a finite sequence $\sigma =(\sigma _0, \ldots , \sigma _m)$ in $V$ such that $\sigma _0=i,\ \sigma _m=j,$ and $\sigma _k\sim \sigma _{k+1}$, for $k=0, \ldots , m-1$. The length of $\sigma$ is defined by $|\sigma |=m$. We denote by $\mathcal{P}_{i,j}^{V}$ the collection of paths in $V$ from $i$ to $j,$ and by $\bar{\mathcal{P}}_{i,j}^{V}$ the collection of paths $\sigma =(\sigma _0=i, \ldots , \sigma _m=j)$ in $V$ from $i$ to $j$ such that $\sigma _k\neq j,k=0,\ldots ,m-1$. For a path $\sigma$ and for $\beta \in {\mathcal{D}}_V^W$, we set

$$\begin{equation} W_\sigma =\prod _{k=0}^{m-1} W_{\sigma _{k},\sigma _{k+1}}, \qquad (2\beta )_\sigma =\prod _{k=0}^{m} (2\beta _{\sigma _{k}}), \qquad (2\beta )^-_\sigma =\prod _{k=0}^{m-1} (2\beta _{\sigma _{k}}). {\tag{3.8}} \end{equation}$$

For the trivial path $\sigma =(\sigma _{0})$, we define $W_{\sigma }=1$, $(2\beta )_{\sigma }=2\beta _{\sigma _{0}}$, $(2\beta _{\sigma })^{-}=1$. (Note that these definitions make sense also in the case of infinite graphs.)

The following representation of the Green function $G(\cdot ,\cdot )$ as a sum on paths will be convenient.

Proposition 6.

Assume that $V$ is finite. For all $\beta \in {\mathcal{D}}_V^W$, we have, with the notations of Theorem A,

$$\begin{equation} G_\beta (i,j)=\sum _{\sigma \in \mathcal{P}_{i,j}^V}\frac{W_{\sigma }}{(2\beta )_{\sigma }}, \;\;\;\; \;\;\;\; \exp (u_\beta (i,j))=\sum _{\sigma \in \bar{\mathcal{P}}_{j,i}^{V}}\frac{W_{\sigma }}{(2\beta )_{\sigma }^-}. {\tag{3.9}} \end{equation}$$

Proof.

Write $D_\beta$ for the diagonal $V\times V$ matrix with $(\beta _i)_{i\in V}$ as diagonal coefficients, then $H_\beta =(\operatorname {Id} -PD_\beta ^{-1} )D_\beta$. Since $H_\beta >0$, by the Perron–Frobenius theorem, we have that $\rho (P D_\beta ^{-1})<1$, where $\rho (P D_\beta ^{-1})$ is the spectral radius of $P D_\beta ^{-1}$. Hence, we can write the following convergent expansion,

$$G_\beta =H_\beta ^{-1}= D_\beta ^{-1}\sum _{k=0}^{\infty } (P D_\beta ^{-1})^k,$$

which exactly corresponds to 3.9.

For the expansion of $\exp (u_\beta (i,j))$, note first that $\sum _{\sigma \in \bar{\mathcal{P}}_{j,i}^{V}}\frac{W_{\sigma }}{\beta _{\sigma }^{-}}\leq \beta _{i}G_\beta (j,i)<\infty$. A path in $\mathcal{P}_{j,i}^{V}$ can be cut at its first visit to $i$, turning it into the concatenation of a path in $\mathcal{\bar{P}}_{j,i}^{V}$ and a path in $\mathcal{P}_{i,i}^{V}$, and this operation is bijective. It implies that

$$\begin{equation} \begin{split} \left(\sum _{\sigma \in \bar{\mathcal{P}}_{j,i}^{V}}\frac{W_{\sigma }}{(2\beta )_{\sigma }^-}\right)G_\beta (i,i)&=\left(\sum _{\sigma \in \bar{\mathcal{P}}_{j,i}^{V}}\frac{W_{\sigma }}{(2\beta )_{\sigma }^-}\right)\left(\sum _{\sigma \in \mathcal{P}_{i,i}^V}\frac{W_{\sigma }}{(2\beta )_{\sigma }}\right)\\ &=\sum _{\sigma \in \mathcal{P}_{j,i}^V}\frac{W_{\sigma }}{(2\beta )_{\sigma }}= G_\beta (i,j), \end{split}{\tag{3.10}} \end{equation}$$

hence the result.

■

3.3. A priori estimates on $\mathcal{Q}_{i_0}^{W}(du)$

The following proposition is borrowed from 10, Lemma 3. For convenience, we give a shorter proof of that estimate based on spanning trees instead of fermionic variables, following the proof of the corresponding result for the ERRW; cf. 8, Lemma 7.

Proposition 7.

Let $(\mathcal{G}=(V,E), W)$ be a finite graph with conductances. Fix a vertex $i_{0}$. Let $\eta >0$ and let $e_1=\{\underline{e_1}, \overline{e_1}\}, \ldots , e_K=\{\underline{e_K}, \overline{e_K}\}$ be $K$ distinct undirected edges such that $W_{e_k}\ge 2\eta$ for all $k=1, \ldots , K$. Then

$$\begin{equation*} \int \exp \left(\eta \sum _{k=1}^{K}\cosh \left(u_{\overline{e_k}}-u_{\underline{e_k}}\right)\right)\mathcal{Q}_{i_0}^{W}(du) \leq e^{\eta K} 2^{K/2}, \end{equation*}$$

where $\mathcal{Q}_{i_0}^{W}(du)$ is the probability distribution defined in Theorem B.

Proof.

Recall that $\mathcal{Q}_{i_0}^{W}(du)$ is defined by

$$\begin{equation*} \mathcal{Q}_{i_0}^{W}(du)= \frac{1}{\sqrt {2\pi }^{|V|-1}}\exp (-\sum _{i}u_i-\sum _{i\sim j}W_{i,j}(\cosh (u_i-u_j)-1))\sqrt {D(W,u)} du_{V\setminus \{i_{0}\}}, \end{equation*}$$

with $du_{V\setminus \{i_{0}\}}=\prod _{i\neq i_0} du_i$ and $D(W,u)=\sum _{T\in {\mathcal{T}}} \prod _{\{i,j\}\in T} W_{i,j}e^{u_i+u_j},$ where the sum is on spanning trees.

Let $\widetilde{W}=W-\eta \sum _{k=1}^{K}\mathds{1}_{e_k}$; i.e., $\widetilde{W}$ is equal to $W-\eta$ on the edges $e_1, \ldots , e_K$ and is unchanged on the other edges. By assumption, we have $\widetilde{W}_{i,j}>0$ on the edges, and for all spanning trees $T$, since edges appear at most once,

$$\begin{align*} \prod _{\{i,j\}\in T}W_{i,j}e^{u_i+u_j}&\leq \left(\prod _{k=1}^K \frac{W_{e_k}}{W_{e_k}-\eta }\right) \prod _{\{i,j\}\in T}\widetilde{W}_{i,j}e^{u_i+u_j} \le 2^K \prod _{\{i,j\}\in T}\widetilde{W}_{i,j}e^{u_i+u_j}, \end{align*}$$

which implies $D(W,u)\le 2^K D(\widetilde{W},u).$ From the expression of $\mathcal{Q}^{W}_{i_0}(du)$, we deduce that

$$\begin{equation*} \exp \left(\eta \sum _{k=1}^{K}\cosh \left(u_{\overline{e_k}}-u_{\underline{e_{k}}}\right)\right) \mathcal{Q}^{W}_{i_0}(du) \le e^{\eta K} 2^{K/2} \mathcal{Q}^{\widetilde{W}}_{i_0}(du). \end{equation*}$$

It implies that

$$\begin{align*} \int \exp \left(\eta \sum _{k=1}^{K}\cosh \left(u_{\overline{e_k}}-u_{\underline{e_k}}\right)\right) \mathcal{Q}^{W}_{i_0}(du) \le e^{\eta K} 2^{K/2} \int \mathcal{Q}^{\widetilde{W}}_{i_0}(du) = e^{\eta K} 2^{K/2}. \end{align*}$$ ■

4. The wired boundary condition and Kolmogorov extension to infinite graphs

4.1. Restriction with wired boundary condition

Our objective is to extend the relations between the VRJP and the $\beta$ field to the case of infinite graphs. To this end, we need an appropriate boundary condition, which turns out to be the wired boundary condition.

Definition 2.

Let $\mathcal{G}=(V,E)$ be a connected graph with finite degree at each site, and let $V_{1}$ be a strict finite subset of $V$. We define the restriction of $\mathcal{G}$ to $V_{1}$ with wired boundary condition as the graph $\mathcal{G}_{1}=(\widetilde{V}_{1}=V_{1}\cup \{\delta \},E_{1})$, where $\delta$ is an extra point and

$$\begin{equation*} E_{1}=\{ \{i,j\}\in E,\ \text{s.t. }i\in V_{1},j\in V_{1},i\sim j\}\cup \{\{i,\delta \}, i\in V_{1} \text{ s.t. } \exists j\notin V_{1},i\sim j\}. \end{equation*}$$

If $(W_{i,j})_{\{i,j\}\in E}$ is a set of positive conductances, we define $(W^{(1)}_{i,j})_{\{i,j\}\in E_{1}}$ as the set of restricted conductances by

$$\begin{equation*} \begin{cases} W_{i,j}^{(1)}=W_{i,j}, & \text{if }i,j\in V_{1}, \;\{i,j\}\in E_1,\\ W_{i,\delta }^{(1)}=\sum _{j\notin V_{1},j\sim i}W_{i,j}, & \text{if } \{i,\delta \}\in E_1,\\ 0,&\text{otherwise.} \end{cases} \end{equation*}$$

Remark 11.

Intuitively, this restriction corresponds to identifying all points in $V\setminus V_1$ to a single point $\delta$ and to deleting the edges connecting points of $V\setminus V_1$. The new weights are obtained by summing the weights of the edges identified by this procedure.

The following lemma is fundamental and is the justification for the choice of this notion of restriction.

Lemma 1.

Let $(\mathcal{G}=(V,E), W)$ be a finite graph with conductances and let $\nu _V^W$ be the associated distribution on random potentials defined in Theorem A. Let $V_1$ be a strict subset of $V$, and let $(\mathcal{G}_1=(\widetilde{V}_1,E_1), W^{(1)})$ be the restriction of $({\mathcal{G}},W)$ to $V_1$ with wired boundary condition. Let $\nu _{\widetilde{V}_1}^{W^{(1)}}$ be the distribution of random potential associated with $({\mathcal{G}}_1, W^{(1)})$. We denote by $\left(\nu _V^W\right)_{|V_1}$ and $\left(\nu _{\widetilde{V}_1}^{W^{(1)}}\right)_{|V_1}$ the marginal distributions on $V_1$ of $\nu _{V}^W$ and $\nu _{\widetilde{V}_1}^{W^{(1)}}$, respectively. Then

$$\begin{equation*} \left(\nu _V^W\right)_{|V_1}=\left(\nu _{\widetilde{V}_1}^{W^{(1)}}\right)_{|V_1}. \end{equation*}$$

Remark 12.

Note that there is no such compatibility relation with the more usual notion of restriction of graph. The wired boundary condition is fundamental and in fact will be responsible for the extra gamma random variable that appears in the representation of the VRJP on the infinite graph.

Proof.

Taking $(\lambda _i)_{i\in V}\in {{\mathbb{R}}}_+^{V}$ such that $\lambda _{V\setminus V_1}=0$ in Theorem A, we get that

$$\begin{equation*} \begin{split} &\int e^{-\sum _{i\in V_1} \lambda _i \beta _i}\nu _V^W(d\beta )\\ &\qquad =\exp \Bigg (-\sum _{i\sim j,i,j\in V_{1}}W_{i,j}(\sqrt {(1+\lambda _{i})(1+\lambda _{j})}-1)\\ &\qquad \qquad \qquad \qquad \quad -\sum _{i\sim j,i\in V_{1},j\notin V_{1}}(W_{i,j}(\sqrt {1+\lambda _{i}}-1))\Bigg )\prod _{i\in V_{1}}\frac{1}{\sqrt {1+\lambda _{i}}}. \end{split} \end{equation*}$$

Applying Theorem A to the graph ${\mathcal{G}}_1$ with $(\lambda _i)_{i\in \widetilde{V}_1}\in {{\mathbb{R}}}_+^{\widetilde{V}_1}$ such that $\lambda _{\delta }=0$, we get

$$\begin{equation*} \begin{split} &\int e^{-\sum _{i\in V_1} \lambda _i \beta _i} \nu _{\widetilde{V}_1}^{W^{(1)}}(d\beta )\\ &\qquad = \exp \Bigg (-\sum _{i\sim j,i,j\in V_{1}}W^{(1)}_{i,j}(\sqrt {(1+\lambda _{i})(1+\lambda _{j})}-1)\\ &\qquad \qquad \qquad \qquad \qquad \quad -\sum _{i\in V_{1}, i{\stackrel{{\mathcal{G}}_1}{\sim }}\delta }(W_{i,\delta }^{(1)}(\sqrt {1+\lambda _{i}}-1))\Bigg )\prod _{i\in V_{1}}\frac{1}{\sqrt {1+\lambda _{i}}}. \end{split} \end{equation*}$$

By the definition of $W^{(1)}_{i,j}$ these Laplace transforms are equal, hence the marginal distributions are equal.

■

4.2. Kolmogorov extension: Proof of Proposition 1 and Definition 1

Let $\mathcal{G}=(V,E)$ be a connected infinite graph with finite degree at each site with conductances $(W_{i,j})$. Recall that $(V_n)_{n\ge 1}$ is an increasing sequence of finite strict subsets of $V$ that exhausts $V$; i.e., $\bigcup _{n} V_n =V.$

Let $\mathcal{G}_n=(\widetilde{V}_{n}=V_n\cup \{\delta _n\}, E_n)$ be the restriction of $\mathcal{G}$ to $V_n$ with wired boundary condition, and let $(W^{(n)})$ be the restricted conductances. By construction if $n<m$, then $({\mathcal{G}}_n, W^{(n)})$ is the restriction of $({\mathcal{G}}_m,W^{(m)})$ to $V_n$ with wired boundary condition. Lemma 1 implies that the sequence of marginal distributions $\left(\nu _{\widetilde{V}_n}^{W^{(n)}}\right)_{|V_n}$ is a compatible sequence of probabilities. By the Kolmogorov extension theorem, it implies that there exists a probability measure $\nu _V^{W}$ such that

$$\left(\nu _V^{W}\right)_{|V_n}= \left(\nu _{\widetilde{V}_n}^{W^{(n)}}\right)_{|V_n},$$

for all integers $n$. By Theorem A, $\nu _V^W(\, \mathrm{d}\beta )$ is supported by the set of potentials $\beta$ such that $(H_\beta )_{V_n,V_n}$ is positive definite for all integers $n$, hence by ${\mathcal{D}}_V^W$. It also implies the other properties of $\nu _V^W(d\beta )$.

The solution of the equation defining $\psi ^{(n)}_\beta$ in Definition 1 exists and is unique since it is equivalent to $(\psi ^{(n)}_\beta )_{V_n^c}=1$ and

$$\begin{align} (H_\beta )_{V_n,V_n}(\psi ^{(n)}_\beta )_{V_n}(i)=\sum _{j\sim i, j\in V_n^c} W_{i,j}, \;\;\; \text{ for }i\in V_n. {\tag{4.1}} \end{align}$$

Since $(H_\beta )_{V_n,V_n}$ is positive definite for $\beta \in {\mathcal{D}}_V^W$, it defines $\psi ^{(n)}_\beta$ uniquely.

4.3. Coupling lemma: Definition of $G^{(n)}$ and relations with $\widehat{G}^{(n)}$, $\psi ^{(n)}$, and $\gamma$

Consider the probability $\nu _V^W(d\beta ,d\gamma )$ defined in 2.5. It will be convenient to couple the measure $\nu _V^W(d\beta ,d\gamma )$ and the measure $\nu _{\widetilde{V}_n}^{W^{(n)}}(d\beta )$ in the following way.

Lemma 2.

For $\beta \in {\mathcal{D}}_V^W$ and $\gamma >0$, we define $\beta ^{(n)}\in {{\mathbb{R}}}^{\widetilde{V}_n}$ by

$$\begin{equation} \beta ^{(n)}_{V_n}=\beta _{V_n},\ \beta ^{(n)}_{\delta _n}= \sum _{j\in V_n, j \sim \delta _n}\frac{1}{2} W^{(n)}_{j,\delta _n} \psi _\beta ^{(n)}(j)+ \gamma . {\tag{4.2}} \end{equation}$$

Then, $\beta ^{(n)}\in {\mathcal{D}}_{\widetilde{V}_n}^{W^{(n)}}$ and under $\nu _V^W(d\beta ,d\gamma )$, $\beta ^{(n)}$ is distributed according to $\nu _{\widetilde{V}_n}^{W^{(n)}}$.

Let $H^{(n)}_{\beta ^{(n)}}$ be the Schrödinger operator associated with ${\mathcal{G}}_n$, $W^{(n)},$ and potential $\beta ^{(n)}$. Let $G_{\beta ^{(n)}}^{(n)}=(H^{(n)}_{\beta ^{(n)}})^{-1}$ be its Green function. Then,

$$G_{\beta ^{(n)}}^{(n)}(\delta _n,\delta _n)={1\over 2\gamma },$$

and, for all $i\in V_n$,

$$\psi ^{(n)}_\beta (i)={G_{\beta ^{(n)}}^{(n)}(\delta _n,i)\over G_{\beta ^{(n)}}^{(n)}(\delta _n,\delta _n)}=e^{u^{(n)}_{\beta ^{(n)}}(\delta _n,i)},$$

where $u^{(n)}_{\beta ^{(n)}}$ is the field defined in Proposition A for the graph ${\mathcal{G}}_n$ and with the potential $\beta ^{(n)}$.

As usual, we often omit the subscript $\beta$ and write $H^{(n)}$, $G^{(n)}$, $u^{(n)}$, and consider them as random variables on ${\mathcal{D}}_V^W\times {{\mathbb{R}}}_+^*$ under $\nu _V^W(d\beta ,d\gamma )$.

Proof.

Let $\beta \in {\mathcal{D}}_V^W$ and $\gamma >0$. Denote in this proof by $(u(j))_{j\in \widetilde{V}_n}$ the vector defined by

$$u(j)= \begin{cases} 0,&\text{ if $j=\delta _n$}, \\ \log \psi _\beta ^{(n)}(j), &\text{ if $j\in V_n$}. \end{cases}$$

Then, by definition of $\psi ^{(n)}_\beta$ and $\beta ^{(n)}$, we have $(H_{\beta ^{(n)}}^{(n)}(e^u))_{V_n}=(H_{\beta }(\psi ^{(n)}_\beta ))_{V_n}=0$ and

$$H_{\beta ^{(n)}}^{(n)}(e^u)(\delta _n)=2\beta ^{(n)}_{\delta _n}- \sum _{j\in V_n, j \sim \delta _n}W^{(n)}_{\delta _n,j} \psi _\beta ^{(n)}(j)=2\gamma .$$

Since $(e^{u(j)})$ is a vector with positive coefficients, by general results on symmetric $M$-matrices; see 19, Theorem 2.7, p. 141, it implies that $H^{(n)}_{\beta ^{(n)}}>0$. Moreover, it implies that ${1\over 2\gamma }e^{u(\cdot )}=G_{\beta ^{(n)}}^{(n)}(\delta _n,\cdot )$, hence that $G_{\beta ^{(n)}}^{(n)}(\delta _n,\delta _n)={1\over 2\gamma }$ and $e^{u(\cdot )}={G_{\beta ^{(n)}}^{(n)}(\delta _n,\cdot )\over G_{\beta ^{(n)}}^{(n)}(\delta _n,\delta _n)}=e^{u_{\beta ^{(n)}}^{(n)}(\delta _n,\cdot )}$.

Finally, by Theorem C, the law of $(\beta ^{(n)}_{V_n}, G^{(n)}_{\beta ^{(n)}}(\delta _n,\delta _n))$ is the same under $\nu _V^W(d\beta ,d\gamma )$ and $\nu _{\widetilde{V}_n}^{W^{(n)}}(d\beta ^{(n)})$, and since $\beta ^{(n)}\mapsto (\beta ^{(n)}_{V_n}, G^{(n)}_{\beta ^{(n)}}(\delta _n,\delta _n))$ is a bijection by Proposition A, it implies that under $\nu _V^W(d\beta ,d\gamma )$, $\beta ^{(n)}$ has law $\nu _{\widetilde{V}_n}^{W^{(n)}}$.

■

Proposition 8.

With the definition of Proposition 2, for all $i,j\in V_n$, all $\beta \in {\mathcal{D}}_V^W$, and all $\gamma >0$,

$$\begin{equation*} {G^{(n)}_{\beta ^{(n)}}(i,j)}= \widehat{G}_\beta ^{(n)}(i,j)+{1\over 2\gamma }\psi _\beta ^{(n)}(i)\psi _\beta ^{(n)}(j). \end{equation*}$$

Proof.

For simplicity, we omit the subscripts $\beta$, $\beta ^{(n)}$ in $\widehat{G}^{(n)}_{\beta }$, $\psi ^{(n)}_\beta$, $G_{\beta ^{(n)}}^{(n)}$ in the expression below. By Proposition 6 and Lemma 2, using $(\beta ^{(n)})_{V_n}=\beta _{V_n}$, we find that

$$\begin{align} G^{(n)}(i,j)=\sum _{\sigma \in \mathcal{P}_{i,j}^{\widetilde{V}_{n}}}\frac{W^{(n)}_{\sigma }}{(2\beta ^{(n)})_{\sigma }} ,\;\;\; \widehat{G}^{(n)}(i,j)= \sum _{\sigma \in \mathcal{P}_{i,j}^{{V}_{n}}}\frac{W_{\sigma }}{(2\beta )_{\sigma }} {\tag{4.3}} \end{align}$$

and

$$\begin{equation*} \psi ^{(n)}(i)=\frac{G^{(n)}(\delta _n,i)}{G^{(n)}(\delta _n,\delta _n)}= \sum _{\sigma \in \overline{\mathcal{P}}_{i,\delta _n}^{\widetilde{V}_{n}}}\frac{W^{(n)}_{\sigma }}{(2\beta )^-_{\sigma }}. \end{equation*}$$

Therefore, if we denote $\mathcal{P}_{i,\delta _n,j}^{\widetilde{V}_{n}}$ the collection of paths on $\widetilde{V}_{n}$ starting from $i$, visiting $\delta _n$ at least once, and ending at $j$, that is,

$$\begin{equation*} \mathcal{P}_{i,\delta _n,j}^{\widetilde{V}_{n}}=\{\sigma =(\sigma _{0},\ldots ,\sigma _{m})\in \mathcal{P}_{i,j}^{\widetilde{V}_{n}}, \text{ such that } \exists 0\leq k\leq m,\sigma _{k}=\delta _n,\}, \end{equation*}$$

then, since $(W^{(n)})_{V_n,V_n}=W_{V_n,V_n}$ and $(\beta ^{(n)})_{V_n}=\beta _{V_n}$,

$$\begin{equation*} \begin{split} &G^{(n)}(i,j)-\widehat{G}^{(n)}(i,j)\\ &\qquad =\sum _{\sigma \in \mathcal{P}_{i,\delta _n,j}^{\widetilde{V}_{n}}}\frac{W^{(n)}_{\sigma }}{(2\beta ^{(n)})_{\sigma }}\\ &\qquad =(\sum _{\sigma \in \bar{\mathcal{P}}_{i,\delta _n}^{\widetilde{V}_{n}}}\frac{W^{(n)}_{\sigma }}{(2\beta ^{(n)})^{-}_{\sigma }})\cdot (\sum _{\sigma \in \mathcal{P}_{\delta _n,j}^{\widetilde{V}_{n}}}\frac{W^{(n)}_{\sigma }}{(2\beta ^{(n)})_{\sigma }})\\ &\qquad =\psi ^{(n)}(i)G^{(n)}(\delta _n,j)=\psi ^{(n)}(i)\psi ^{(n)}(j)G^{(n)}(\delta _n,\delta _n)= \psi ^{(n)}(i)\psi ^{(n)}(j){1\over 2\gamma }, \end{split} \end{equation*}$$

where we used Lemma 2 in the last equality.

■

5. The martingale property

Recall that ${{\mathcal{F}}}^{(n)}=\sigma (\beta _i, i\in V_n)$ is the sub-$\sigma$-field generated by the random variables $\beta \mapsto \beta _i$, $i\in V_n$. The following proposition is the key property for the main theorem.

Proposition 9.

With the notations of Definition 1, for all $n\in {{\mathbb{N}}}$, $\psi ^{(n)}$ has finite moments. Moreover, we have, $\nu _V^W$-a.s.,

$$\begin{equation} {{\mathbb{E}}}_{\nu _V^W}\left(\psi ^{(n+1)}(i)| {{\mathcal{F}}}^{(n)}\right)= \psi ^{(n)}(i) \;\;\; \forall i\in V, {\tag{5.1}} \end{equation}$$

and for all $i,j\in V$,

$$\begin{equation} \begin{split} {{\mathbb{E}}}_{\nu _V^W}&\left(\psi ^{(n+1)}(i)\psi ^{(n+1)}(j)-\psi ^{(n)}(i)\psi ^{(n)}(j) | {{\mathcal{F}}}^{(n)}\right)\\ &= {{\mathbb{E}}}_{\nu _V^W}\left(\widehat{G}^{(n+1)}(i,j)-\widehat{G}^{(n)}(i,j)| {{\mathcal{F}}}^{(n)}\right). \end{split}{\tag{5.2}} \end{equation}$$

Remark 13.

In Theorem B, by the change of variables $\widetilde{u}(\cdot )=u(\cdot )-\frac{\sum _{i\in V}u(i)}{|V|}$, the new variables $(\widetilde{u}(i))_{i\in V}$ are in the space $\{\sum _{i\in V}\widetilde{u}(i)=0\}$ and the density becomes

$$\begin{equation*} \mathcal{\widetilde{Q}}_{i_{0}}^{W}(d \widetilde{u})=\frac{1}{\sqrt {2}^{|V|-1}}e^{\widetilde{u}(i_{0})}e^{-\sum _{i\sim j}W_{i,j}(\cosh (\widetilde{u}(i)-\widetilde{u}(j))-1)}\sqrt {D(W,\widetilde{u})} d \widetilde{u}_{V\setminus \{i_{0}\}}. \end{equation*}$$

We see from this expression that $e^{\widetilde{u}(i)-\widetilde{u}(i_0)}\cdot \mathcal{\widetilde{Q}}_{i_{0}}^{W}=\mathcal{\widetilde{Q}}_{i}^{W}$, and hence that

$$\begin{equation*} \int e^{\widetilde{u}(i)-\widetilde{u}(i_0)}\mathcal{\widetilde{Q}}_{i_{0}}^{W}(d \widetilde{u})=1. \end{equation*}$$

Applied to $V=\widetilde{V}_{n}$, $i_{0}=\delta _{n}$, we get ${{\mathbb{E}}}_{\nu _V^W}(\psi ^{(n)}(i))=1,$ which is a particular case of 5.1.

The original proof of that property was rather technical (see the second arXiv version of the present paper). Some time after the first version of this paper was posted on arXiv, a simpler proof of the martingale property 5.1 was given in 7. Moreover, using some supersymmetric arguments, the following more general property was proved.

Lemma 3 (7).

Let $\lambda \in ({{\mathbb{R}}}_+)^V$ be a nonnegative function on $V$ with bounded support. Then

$${{\mathbb{E}}}\left(e^{-\left<\lambda ,\psi ^{(n+1)}\right>-\frac{1}{2}\left<\lambda , \widehat{G}^{(n+1)}\lambda \right>} | {{\mathcal{F}}}^{(n)}\right)=e^{-\left<\lambda ,\psi ^{(n)}\right>-\frac{1}{2}\left<\lambda , \widehat{G}^{(n)}\lambda \right>}.$$

We provide here a different proof of this assertion based on elementary computations on the measures $\nu _V^W$ on finite sets. It also provides a simpler proof of the original assertion Proposition 9 by differentiating in $\lambda$.

5.1. Marginal and conditional laws of $\nu _V^{W}$

In this subsection we suppose that $\mathcal{G}=(V,E)$ is finite. We state some identities on marginal and conditional laws of the distribution $\nu _V^{W}$, which will be instrumental in the proof of the martingale property in the next subsection.

Let us first remark that the law $\nu _V^{W}$ defined in Theorem A can be extended to the case where $P=(W_{i,j})_{i,j\in V}$ has nonzero, diagonal coefficients. Indeed, if some diagonal coefficients of $P$ are positive, then changing from variables $(\beta _i)$ to variables $(\beta _i-\frac{1}{2}W_{i,i})$, we get the law $\nu _V^{\widetilde{W}}$ where $(\widetilde{W}_{i,j})$ is obtained from $(W_{i,j})$ by replacing all diagonal entries by 0. While it is not very natural from the point of view of the VRJP to allow nonzero diagonal coefficients, it is convenient in this section to allow this possibility since it simplifies the statements about conditional law.

Recall that for any function $\zeta :V\mapsto {{\mathbb{R}}}$ and any subset $U\subset V$, we write $\zeta _U$ for the restriction of $\zeta$ to the subset $U$. Similarly, if $A$ is a $V\times V$ matrix and $U\subset V$, $U'\subset V$, we write $A_{U,U'}$ for its restriction to the block $U\times U'$. We also write $d\beta _{U}=\prod _{i\in U}d\beta _{i}$ to denote integration on variables $\beta _U$.

In the next lemma we give an extension of the family of probability distributions $\nu _V^{W}$. This extension was proposed by Letac, in the unpublished note 13 discussing the integral defined in 22. We give a proof of this lemma using Theorem A.

Lemma 4 (Letac, 13).

Let $V$ be finite, and let $P=(W_{i,j})_{i,j\in V}$ be a symmetric matrix with nonnegative coefficients. Let $(\eta _i)_{i\in V}\in {{\mathbb{R}}}_+^V$ be a vector with nonnegative coefficients. Then the following measure on ${\mathcal{D}}_V^W$

$$\begin{align} \nu _V^{W,\eta }(d\beta ) &:=e^{-\frac{1}{2}\left<\eta , (H_\beta )^{-1} \eta \right>}e^{\left<\eta ,1\right>} \nu _V^{W}(d\beta ) {\tag{5.3}}\\ &=\mathds{1}_{H_{\beta }>0}\left(\frac{2}{\pi }\right)^{|V|/2}e^{-\frac{1}{2}\left<1, H_\beta 1\right>-\frac{1}{2}\left<\eta , (H_\beta )^{-1} \eta \right> }\frac{1}{\sqrt {\det H_{\beta }}} e^{\left<\eta ,1\right>} d\beta _V \end{align}$$

is a probability distribution, where $1$ in the scalar products $\left< 1,H_{\beta }1 \right>$ and $\left< \eta , 1 \right>$ is to be understood as the vector $\begin{pmatrix} 1\\ \vdots \\ 1 \end{pmatrix}$. Its Laplace transform is, for any $\lambda \in \mathbb{R}_{+}^{V}$,

$$\begin{equation} \int e^{-\left< \lambda ,\beta \right>} \nu ^{W,\eta }_V(d\beta )\!=\!e^{-\left< \eta ,\sqrt {\lambda +1}-1 \right>-\sum _{i\sim j}W_{i,j}\left( \sqrt {(1+\lambda _{i})(1+\lambda _{j})}-1 \right)}\!\prod _{i\in V}\frac{1}{\sqrt {1+\lambda _{i}}}, {\tag{5.4}} \end{equation}$$

where $\sqrt {\lambda +1}-1$ should be considered as the vector $(\sqrt {\lambda _i+1}-1)_{i\in V}$.

It appears in the following lemma that this extension describes all marginal laws of $\nu _V^{W}$ and that the larger family $\nu _V^{W,\eta }$ is stable by taking marginals and conditional distributions.

Lemma 5.

Assume that $V$ is finite, and let $U\subset V$ be a subset. Under $\nu _V^{W,\eta }(d\beta )$,

(i)

$\beta _{U}$ is distributed according to $\nu _{U}^{W_{U,U},\widehat{\eta }},$ where$$\begin{align} \widehat{\eta }= \eta _{U}+ P_{U,U^c} (1_{U^c}), {\tag{5.5}} \end{align}$$

(ii)

conditionally on $\beta _{U}$, $\beta _{U^c}$ is distributed according to $\nu _{U^c}^{\widecheck W,\widecheck \eta }$, where $\widecheck P=(\widecheck W_{i,j})_{i,j\in U^c}$ and $\widecheck \eta \in ({{\mathbb{R}}}_+)^{U^c}$ are the matrix and vector defined by$$\begin{equation*} \widecheck P=P_{U^c,U^c}+ P_{U^c,U} \left((H_\beta )_{U,U}\right)^{-1} P_{U,U^c}, \;\;\; \widecheck \eta =\eta _{U^c}+P_{U^c,U} \left((H_\beta )_{U,U}\right)^{-1}(\eta _{U}). \end{equation*}$$

Remark 14.

Note that $\widecheck P$ has nonzero diagonal coefficients.

N.B.

As we can observe, all the quantities with $\widecheck \cdot$ are relative to vectors or matrices on $U^{c}$, while the quantities with $\widehat{\cdot }$ are relative to vectors or matrices on $U$.

Lemma 6.

Let $\mathcal{G}=(V,E)$ be a finite connected graph endowed with conductances $P=(W_{i,j})_{i,j\in V}$. Let $(\eta _i)_{i\in V}\in {{\mathbb{R}}}_+^V$ be a vector with nonnegative coefficients. Let $U\subset V$. For $\beta \in {\mathcal{D}}_V^W$, define $\psi _\beta =G_{\beta }\eta ,$ where $G_{\beta }=H_{\beta }^{-1}$, and define $\widehat{\eta }=\eta _{U}+P_{U,U^{c}}1_{U^{c}}$, $\widehat{G}_\beta ^{U}=((H_{\beta })_{U,U})^{-1}$ and $\widehat{\psi }_\beta =\widehat{G}_\beta ^{U}(\widehat{\eta })$. For any $\lambda \in \mathbb{R}_{+}^{V}$, we have, $\nu _V^{W,\eta }$ a.s.,

$$\begin{equation} {{\mathbb{E}}}_{\nu _{V}^{W,\eta }}(e^{-\left< \lambda ,\psi \right>-\frac{1}{2}\left< \lambda ,G \lambda \right>}\left| \mathcal{F}_{U}\right.)=e^{-\left< \lambda _{U},\widehat{\psi } \right>-\left< \lambda _{U^c},1_{U^c} \right>-\frac{1}{2}\left< \lambda _{U},\widehat{G}^{U}\lambda _{U} \right>}, {\tag{5.6}} \end{equation}$$

where $\mathcal{F}_{U}=\sigma (\beta _{i},i\in U)$.

Proof of Lemmas 4 and 5.

Lemma 4 and the assertions (i) and (ii) of Lemma 5 are consequences of the same decomposition of the measure $\nu _V^{W,\eta }$. It is partially inspired by computations in 13. We write $H_{\beta }$ as a block matrix,

$$\begin{equation*} H_{\beta }= \begin{pmatrix} H_{U,U} & -P_{U,U^{c}} \\ -P_{U^{c},U} & H_{U^{c},U^{c}} \end{pmatrix} \text{ and define }\widehat{G}^{U}=(H_{U,U})^{-1}. \end{equation*}$$

Now, define the Schur’s complement

$$\begin{align} \widecheck H^{U^{c}}=H_{U^{c},U^{c}}-P_{U^{c},U}\widehat{G}^{U}P_{U,U^{c}} {\tag{5.7}} \end{align}$$

and

$$\begin{align*} \widecheck G^{U^c}=\left( \widecheck H^{U^{c}}\right)^{-1}. \end{align*}$$

We have

$$\begin{equation} H_{\beta }= \begin{pmatrix} I_{U} & 0 \\ -P_{U^{c},U}\widehat{G}^{U} & I_{U^{c}} \end{pmatrix} \begin{pmatrix} H_{U,U} & 0 \\ 0 & \widecheck H^{U^{c}} \end{pmatrix} \begin{pmatrix} I_{U} & -\widehat{G}^{U}P_{U,U^{c}} \\ 0 & I_{U^{c}} \end{pmatrix}. {\tag{5.8}} \end{equation}$$

We remark that with notations of Lemma 5(ii), we have

$$\widecheck H^{U^c}=2\beta _{U^c}-\widecheck P.$$

By 5.8, we have

$$\begin{equation} \begin{split} &\left< 1,H_{\beta }1 \right> \\ &\qquad = \left< 1_{U^c},\widecheck {H}^{U^c}1_{U^c} \right>+\left< 1_{U},H_{U,U}1_{U} \right>\\ &\qquad \quad + \left< 1_{U^c},P_{U^{c},U}\widehat{G}^{U} P_{U,U^{c}} 1_{U^c}\right>-2\left< 1_{U},P_{U,U^{c}} 1_{U^c}\right>. \end{split} {\tag{5.9}} \end{equation}$$

On the other hand, by 5.8 again, we have

$$\begin{equation} G_{\beta }= H_\beta ^{-1}= \begin{pmatrix} I_{U} & \widehat{G}^{U}P_{U,U^{c}} \\ 0 & I_{U^{c}} \end{pmatrix} \begin{pmatrix} \widehat{G}^{U} & 0\\ 0 & \widecheck {G}^{U^c} \end{pmatrix} \begin{pmatrix} I_{U} & 0 \\ P_{U^{c},U}\widehat{G}^{U} & I_{U^{c}} \end{pmatrix}, {\tag{5.10}} \end{equation}$$

therefore, since

$$\begin{equation*} \begin{pmatrix} I_{U} & 0 \\ P_{U^{c},U}\widehat{G}^{U} & I_{U^{c}} \end{pmatrix} \begin{pmatrix} \eta _{U} \\ \eta _{U^{c}} \end{pmatrix}= \begin{pmatrix} \eta _{U}\\ \widecheck {\eta } \end{pmatrix} , \end{equation*}$$

we get

$$\begin{equation} \left< \eta ,G_{\beta }\eta \right>=\left< \eta _{U},\widehat{G}^{U}\eta _{U} \right>+\left< \widecheck \eta , \widecheck {G}^{U^c} \widecheck \eta \right>. {\tag{5.11}} \end{equation}$$

Combining 5.9 and 5.11, we have

$$\begin{align} \left< 1,H_{\beta }1 \right>+\left< \eta ,G_{\beta }\eta \right>-2\left<\eta ,1\right> =&\left< 1_{U^c},\widecheck {H}^{U^c}1_{U^c} \right>+ \left< \widecheck {\eta },\widecheck {G}^{U^c} \widecheck {\eta } \right>-2\left<\widecheck \eta ,1_{U^c}\right>{\tag{5.12}}\\ &+\left< 1_{U},H_{U,U}1_{U} \right>+\left< \widehat{\eta },\widehat{G}^{U}\widehat{\eta }\right>-2\left<\widehat{\eta },1_{U}\right>. \end{align}$$

By 5.8 we also have

$$\begin{align} \det H_{\beta }=\det H_{U,U} \det \widecheck {H}^{U^c},\ \ \mathds{1}_{H_{\beta }>0}=\mathds{1}_{H_{U,U}>0}\mathds{1}_{\widecheck {H}^{U^c}>0}. {\tag{5.13}} \end{align}$$

Combining 5.12 and 5.13, we have

$$\begin{align} & \left( \frac{2}{\pi } \right)^{|V|/2} e^{-\frac{1}{2}\left< 1,H_{\beta }1 \right>-\frac{1}{2}\left< \eta ,G_{\beta }\eta \right>+\left<\eta ,1\right>}\frac{\mathds{1}_{H_{\beta }>0}}{\sqrt {\det H_{\beta }}} \\ &=\left( \frac{2}{\pi } \right)^{|U|/2} e^{-\frac{1}{2} \left< 1_U,H_{U,U}1_U \right>-\frac{1}{2}\left<\widehat{\eta },\widehat{G}^{U}\widehat{\eta }\right>+\left<\widehat{\eta },1_{U} \right> }\frac{\mathds{1}_{H_{U,U}>0}}{\det H_{U,U}} {\tag{5.14}}\\ &\;\;\;\;\cdot \left( \frac{2}{\pi } \right)^{|U^{c}|/2} e^{-\frac{1}{2}\left< 1_{U^c},\widecheck {H}^{U^c}1_{U^c} \right> -\frac{1}{2}\left< \widecheck {\eta },\widecheck {G}^{U^c} \widecheck {\eta } \right>+\left<\widecheck \eta ,1_{U^c}\right>}\frac{\mathds{1}_{ \widecheck {H}^{U^c}>0}}{\sqrt {\det \widecheck {H}^{U^c}}} . \end{align}$$

We remark that the left-hand side is the density of $\nu _V^{W,\eta }(d\beta )$, that the first term of the right-hand side corresponds to the density of $\nu _{U}^{W_{U,U},\widehat{\eta }}(d\beta _U),$ and that, $\beta _U$ being fixed, the second term of the right-hand side is the density of $\nu _{U^c}^{\widecheck W,\widecheck \eta }(d\beta _{U^c})$. (Indeed, as remarked above, $\widecheck H^{U^c}=2\beta _{U^c}-\widecheck P$ and $\widecheck P$, $\widecheck \eta$ are $\beta _U$-measurable.)

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Proof of Lemma 4.

Take $\eta =0$. Then $\widecheck \eta =0$. Integrating on $d\beta _{U^c}$ on both sides of 5.14, with $\beta _{U}$ fixed, gives