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# Random Metric Geometries on the Plane and Kardar-Parisi-Zhang Universality

Communicated by *Notices* Associate Editor Scott Sheffield

*A central object featuring in a significant aspect of modern research in probability theory is a random metric space obtained by distorting Euclidean space using some form of random noise. A topic of fundamental importance in this regard is then how the geodesics behave in the presence of noise.*

While examples of random metric spaces occur naturally in various physical situations, a few things have to be made precise before formulating concrete mathematical questions. For simplicity, we will work in a “discrete” setting, i.e., instead of consider the lattice , with the usual nearest neighbor graph metric. In the latter, the distance between two points is the smallest number of edges on any path connecting the same, i.e., this is simply the well-known metric.

We can now describe a canonical model of random geometry introduced in 1965 by Hammersley and Welsh. The description is deceptively simple! For each edge let , be an i.i.d. (independent and identically distributed) copy of a non-negative random variable with a fixed distribution, say Thus, the “edge weight” . can be thought of as the random length of the edge which was previously simply one, and this leads to a new (random) shortest path metric on , Let . and denote the random distance and the corresponding shortest path(s) between and respectively. ,

Originally this was suggested as a model of fluid flow through a porous but random medium where for any edge its weight denotes the time taken by the fluid to cross it, often termed as the *passage time*. Hence the model is called First Passage Percolation (FPP). Subsequently this has been of interest in many areas of physics and biology as well as computer science, modeling various natural phenomena including spreading of bacteria and infection, propagation of forest fires and flame fronts, flow of current, and more.

It might be instructive to begin by listing a series of questions for FPP that one might wish to answer (for several more, see ADH17).

- (1)
*How does the metric ball centered at any point, say i.e., , grow ,*?- (2)
*The above question can be broken down into two broad subparts. The first is about comparing the macroscopic behavior to the metric on How does . behave compared to*- (3)
*The next question is more refined and is about the order of fluctuations. How does ( denotes the expectation) behave? More generally, how does the boundary of the metric ball fluctuate?*- (4)
*Moving on from the next thing to investigate is , the geodesic(s) between , and What is its fluctuation? In particular, how much does it deviate from the straight line joining . and ?*- (5)
*Finally, is there any sense in which one can take the limit of the***entire**metric space? Is the limit universal, i.e., it does not depend on the edge weight distribution as long as it is “sufficiently nice”? ,

Perhaps a bit surprisingly, it turns out that this simple mathematical model is notoriously hard to analyze with most of the above questions in full generality still remaining open. Nonetheless, impressive, though somewhat limited, advances have been made over the years giving birth to key new ideas and techniques in mathematics, statistical physics, and probability theory. Recent years have also seen an explosion of activity around related models of random geometry admitting certain “exact formulas.”

In this article we will review some of the general ideas, followed by the more recent developments around the specialized exactly solvable examples.

## 1. Limiting Shape and Fluctuations

Starting with questions 1 and 2 about the growth rate of the metric, the classical law of large numbers result (LLN) states that for any i.i.d. sequence of random variables , … with finite mean, i.e., , the long-term average of the sequence converges to , i.e., , This in particular implies that the sum of i.i.d. variables grows linearly if . .

Given the above, it is natural to wonder if a counterpart result exists for FPP where the random metric is also driven by i.i.d. variables, albeit in a complicated way. This was answered by Richardson and further refined by Cox, Durrett, and Kesten in the form of the following *shape theorem* (see Figure 1).

Thus, in words, the above says that the metric ball around the origin (or for that matter around any point) essentially looks like a dilated version (by a factor of of ) In particular it says that for each unit . the passage time has an asymptotic speed , depending on the geometry of i.e., , This answers the question about growth rate in full generality. Note that this is a .*non-universal* result, as the limiting shape depends on the details of the edge weight distribution although, perhaps surprisingly, the precise description is not explicitly known in any case. ,

We now arrive at question 3, about fluctuations, i.e., say for how does ,

behave? This seemingly innocuous question is one of the key problems in probability theory!

To begin, let us start with a classical fluctuation counterpart of the above law of large numbers result. Under the additional assumption (where denotes the variance), the well-known central limit theorem (CLT) states that

where is a standard Gaussian random variable and the convergence is in distribution. Note that unlike the LLN result, the above is *universal*, yielding the Gaussian distribution in the limit regardless of the distribution of the variables .

Beyond weak convergence, often in applications, it is useful to obtain concentration bounds of the following kind (which need further assumptions that we will not spell out). There exists a universal constant such that for all

Concentration of measure for linear functions of i.i.d. variables as above is very well understood. However in many natural examples, such as FPP, the observables of interest are complicated non-linear functions of i.i.d. variables. Proving concentration bounds for such random variables poses a major challenge in probability theory with diverse applications. Over the years the theory has seen major advances. In the context of FPP, concentration results have been proven in landmark results of Kesten followed by Talagrand, further by Benjamini, Kalai, Schramm, and then subsequently improved across various works (see e.g., ADH17). Below we record a somewhat informal statement which captures the flavor of some of these results.

Note that the above concentration result for

Towards this, observe that while the Gaussian tail bound immediately implies an order *hypercontractivity*, and *is the best known bound in any generality currently!* More recently, corresponding improvements of 1.2 at scale

Nonetheless, using non-rigorous methods one can in fact predict that

## 2. Kardar-Parisi-Zhang Universality

As indicated in the discussion following Theorem 1.1, while the growth rate is non-universal in the shape theorem, the fluctuation theory is expected to be universal and one might wonder what the counterpart object for FPP is, analogous to the Gaussian distribution appearing in the CLT. It turns out, in dimension 2 (recall we have been considering FPP on

To begin, we first introduce another model of stochastic growth which is closely related to FPP, but with some key differences, and with a similar name, Last Passage Percolation (LPP). Although expected to behave similarly, it turns out that the latter exhibits some extremely useful and surprising properties absent in FPP, which will be apparent soon.

**Last Passage Percolation.** Consider the lattice upper half-plane *oriented* paths from

This slight alteration of the setting, i.e., considering oriented paths and maximum passage times, along with certain special choices of the vertex weights, will lead to quite remarkable algebraic properties which form the central underpinning of most of the significant advances in our understanding of such examples. Going forward, abusing terminology, we will call the maximizing path(s) the geodesic(s) between

As in FPP, one considers the growing cluster around any point, say the origin (to be denoted throughout by

On simulating either FPP or LPP on the computer, one observes that the boundary of the growing cluster around the origin exhibits certain key features (it will be useful to view 2 dimensions as one “spatial” direction and one “time” direction; see Figure 2). This includes a global smoothening phenomenon involving faster growth in rougher portions on the boundary than smoother parts. Further, crucially, the growth rate is expected to depend non-linearly on the local gradient. Finally, as is obvious from the descriptions of the models, the local fluctuations are driven by i.i.d. noise.

In 1986, physicists Kardar, Parisi, and Zhang (KPZ) KPZ86 put forward a unified fluctuation theory predicting that the behavior of stochastic growth exhibiting the above features, e.g., the boundary of the growth cluster in LPP, should be the same as that of a canonical non-linear stochastic PDE (the KPZ equation). For more on the KPZ equation, see, e.g., Qua11Cor16.

While several models are expected to be in the KPZ universality class, keeping with our theme of random metric spaces and geometry of geodesics, we will simply be focussing on FPP and LPP throughout the article, as they enjoy special geometric properties missing in the other examples.

### 2.1. Characteristic exponents governing fluctuation.

Much of the predictions about the fluctuation theory for such models can be summarized in the triple of “critical” exponents

Answering this and explaining the

confirming the intuition that this is much more concentrated than the weight of any given path joining the points, which has

The exponent

In particular,

However, despite these intriguing predictions being around for multiple decades, the rigorous literature is lagging behind significantly as we have seen in FPP, where the best known bound on fluctuations is still

Nonetheless, while the general situation is admittedly somewhat disappointing, a handful of examples possess certain special properties that have opened the door to various external mathematical tools leading to spectacular progress over the last twenty years. This brings us into the world of integrable probability.

## 3. Integrable Probability

A class of models remarkably exhibit additional algebraic structure, though often quite difficult to discern, which can be exploited to obtain exact expressions for the observables of interest. The formulas can then be analyzed to extract information about their asymptotic behavior. These include connections to representation theory, algebraic combinatorics, and in particular to permutations, random matrices, and their eigenvalues, quantum integrable systems, and so on. To convey the basic idea, we will discuss a particular example of how a certain special choice of the vertex weights in LPP makes it *integrable* or *exactly solvable*. For more examples, see BG16.

• **Exponential LPP.** This is a model of LPP when the vertex weights

To describe a key consequence of the algebraic properties of this model, we need to recall a classical random matrix ensemble.

• **Laguerre Unitary Ensemble (LUE).** For any positive integers

We now discuss Johansson’s breakthrough observation: Exponential LPP is related to LUE and this can be used to verify KPZ predictions!

### 3.1.
exponent guiding weight fluctuations via random matrix theory

The following remarkable identity for Exponential LPP was established by Johansson Joh00:

where

where

Thus this surprising turn of events connecting random planar growth and random matrices not only rigorously establishes the predicted

The first such result was proven by Baik, Deift, and Johansson in the context of a different model of LPP which is naturally connected to the well-known problem of the longest increasing subsequence in a random permutation formulated by Ulam. We point the reader to the beautiful book Rom15 discussing in depth the above problem.

Having established the

### 3.2.
exponent and spatial correlation structure

Continuing from Section 2.1, to understand the correlation structure of *at scale * it will be convenient to consider the scaled

*geodesic weight profile*,

which encodes the properly centered and scaled last passage times from

Again, it might be instructive to draw analogy with the classical case of a sequence of i.i.d. variables. Towards this, we recall Donsker’s invariance principle, a process version of the CLT result. Namely, if one considers the random function

for

However, we have already seen from 3.2 that *Parabolic Airy*, which we will denote as (see Figure 4)

This is the central object in the KPZ universality class expected to arise as the universal scaling limit of fluctuations in a large class of stochastic planar growth models including random metric spaces!

Here, *determinantal* which admits usage of algebraic tools. Without dwelling much on this, we simply mention that the latter means that the joint distribution of the values taken at different points is given by determinants of certain matrices. This in particular tells us how the correlation of the geodesic weight behaves at scale

The Airy

Finally, despite all the above acting as evidence that the first naive guess of every scaling limit being Brownian motion does not hold in this case, as we will now see, it wasn’t too far!

### 3.3. Local Brownianity of the Airy process

Though the Airy

The first result in this direction was proven by Hägg. Subsequently, Corwin and Hammond showed that for the Parabolic Airy*indistinguishable* from a single sample of a properly scaled Brownian motion (see Figure 4). However one cannot hope to have such an indistinguishability result on the whole real line since Airy

The key tool in the proof? Again remarkable connections to random matrices. The argument in the Corwin-Hammond work relied on a gorgeous identity proved by O’Connell-Yor OY02 which states that for Brownian LPP (a variant of Exponential LPP, where instead of sums of exponential variables one has Brownian motions) the geodesic weight profile, also known to converge to

As an immediate application, since the same is true for Brownian motion, one concludes that the Airy

Thus, for certain models of LPP we have seen that the geodesic weight profile, scaled using KPZ exponents, converges to the Parabolic Airy

As the reader might have noticed, in all the results outlined so far, connections to random matrices play a major role, and these highly non-obvious features only exist because of certain special choices made while defining the models, such as using exponential variables or Brownian motions. There are other cases which are not directly connected to matrices but to other objects such as permutations but in the interest of brevity we will not be reviewing them and instead point the reader to BG16.

We now come to question 5 about the scaling limit of the metric space itself. Towards this, so far we have seen that when one point is fixed to be the origin, the distance to another point as the latter varies along a straight horizontal line scales to

### 3.4. The Directed Landscape

To talk about such scaling limits, we need to first upgrade our notation

which encodes the scaled geodesic weight between scaled points

In 2018, Dauvergne, Ortmann, and Virag DOV constructed the Directed Landscape: a four-parameter random energy field

Having obtained the Directed Landscape

• *Translation invariance*:

• *Invariance under KPZ *:

• *Independence of increments*: for any time points

Note that

## 4. Geodesic Geometry and Its Consequences

Recall that the article started by indicating that one of its main motivations was to study geodesics in random distortions of the Euclidean metric. All of the previous preparation now allows us to dive into geodesic geometry in integrable models of LPP, an area that has seen an explosion of activity recently. We start by recording some basic but fundamental properties.

• **Transversal fluctuations.** Recall that the transversal fluctuation of the geodesic from the straight line was mentioned to be governed by the exponent

and

• **Coalescence of geodesics.** We now come to perhaps the most striking feature of the random metric spaces being discussed, standing sharply in contrast to Euclidean geometry. This is the phenomenon of *coalescence*. As the name suggests, this means that geodesics between pairs of points that are not very far away from each other tend to meet and share a non-trivial amount of their journey (see, e.g., the figure on the first page which illustrates a network of geodesics!). This is very different from Euclidean geometry where geodesics are straight lines and meet at a single point or not at all. In very brief, the reason for the above is that geodesics are weight maximizing paths, and hence each path, regardless of its endpoints, prefers to pass through the vertices whose random weights are the highest, leading them to meet each other, i.e., coalesce.

While the initial progress in understanding properties of random antimetrics arising in LPP, such as their fluctuations, was mostly through algebraic methods, the past few years have seen several advances with a focus on geodesic behavior, based primarily on geometric and probabilistic analysis. While it is impossible to review all of them, we include a sample list below to provide the reader a flavor.

### 4.1. Applications

• **Fractal geometry.** Fractals are self-similar sets; they look the same at all scales. They are ubiquitous in nature and in mathematics, with the famous Cantor set being a classical example.

Random fractal sets are also commonly occurring in probability. A good way is to think of them as rather sparse random subsets of

The scale invariance satisfied by the Directed Landscape makes it a rich source of intricate fractal behavior. We now describe a particular example related to the behavior of geodesics. It turns out that for any two fixed points *disjoint* geodesics between the points

• **Temporal correlation.** It is of much interest to understand how values at different points in space and time in the Directed Landscape are correlated. While determinantal formulas have provided a reasonable understanding of the spatial correlation observables such as

• **Non-existence of bigeodesics.** As we have seen, geodesics behave very differently in these types of random metric spaces compared to Euclidean spaces. Another related notion in which they are expected to differ pertains to the existence of bigeodesics, i.e., a bi-infinite path such that every finite segment of it is a geodesic. Note that any straight line is a bigeodesic in Euclidean space, whereas a central problem in FPP on