Local Semicircle Law in the Bulk for Gaussian β-Ensemble

Abstract

We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of n ^−1+δ for any δ>0. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of n ^−1/2+δ; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erdös, and Yau in arXiv:1104.2272 [math.PR] and Bao and Su in arXiv:1104.3431 [math.PR]. Our approach is new and could be extended to other tridiagonal models.

Appendix

Here we prove a slight modification of McDiarmid’s inequality used in our inductive argument.

Proposition A.7

A modification of McDiarmid’s inequality

Let X ₁,…,X _n be independent subgaussian random variables and suppose F is a function of n variables such that there exists an event Ω with overwhelming probability that if $x_{1},\ldots ,x_{n},\tilde{x_{i}} \in\varOmega$, then

$$ \bigl|F(x_1,\ldots,x_n) - F(x_1, \ldots,x_{i-1},\tilde{x_i},x_{i+1}, \ldots,x_n)\bigr| \leq c_i $$

for all 1≤i≤n, and outside of Ω, F is bounded by a polynomial of n. Then for any λ>0, one has

$$ \mathbb{P}\bigl(\bigl|F(X)-\mathbb{E}\bigl(F(X)\bigr)\bigr| \geq\lambda\sigma \bigr) \leq C \exp\bigl(-c\lambda^2\bigr) $$

for some absolute constant C,c>0, and $\sigma= \sum_{i=1}^{n}c_{i}^{2}$.

Proof

By symmetry, it suffices to show

$$ \mathbb{P}\bigl(F(X)-\mathbb{E}F(X) \geq\lambda\sigma\bigr) \leq C\exp \bigl(c\lambda^2\bigr). $$

Let t>0 be a parameter to be chosen later. Consider the exponential moment

$$ \mathbb{E}\bigl(\exp\bigl(tF(X)\bigr)\bigm|X_1,\ldots,X_{n-1},\ \{X_i\}\in\varOmega\bigr). $$

Writing $Y = F(X) - \mathbb{E}(F(X)\mid X_{1},\ldots,X_{n-1},\ \{X_{i}\}\in \varOmega)$, we can rewrite the above as:

$$ \mathbb{E}\bigl(\exp\bigl(tF(Y)\bigr)\bigm|X_1,\ldots,X_{n-1},\ {X_i}\in\varOmega\bigr) \exp\bigl(t\mathbb{E}\bigl(F(X) \bigr)\bigm|X_1,\ldots,X_{n-1},\ \{X_i\}\in\varOmega\bigr). $$

By the condition of the theorem, tY fluctuates only by at most tc _n and has mean 0. By Hoeffding’s lemma, we have

$$ \mathbb{E}\bigl(\exp\bigl(tF(Y)\bigr)\bigm|X_1,\ldots,X_{n-1},\ \{X_i\}\in\varOmega\bigr) \leq\exp\bigl(O\bigl(t^2c_n^2\bigr)\bigr). $$

Integrating out the conditioning, we have the bound

$$ \mathbb{E}\bigl(\exp\bigl(tF(X)\bigr) \leq\exp\bigl(O\bigl(t^2c_n^2 \bigr)\bigr)\bigr)\mathbb{E}\bigl(t\bigl(\mathbb{E}F(X)\bigm|X_1, \ldots,X_{n-1},\ \{X_i\}\in\varOmega\bigr)\bigr). $$

Now the latter expectation is a function of the first (n−1) variables and obeys the same hypothesis, so we can iterate and obtain the bound:

$$ \exp\Biggl(\,\sum_{i=1}^nO \bigl(t^2c_i^2\bigr)\Biggr) \mathbb{E} \bigl(t\bigl(\mathbb{E}F(X)\bigm|\{X_i\}\in\varOmega\bigr)\bigr) $$

and by the overwhelming probability condition and the bound outside of the set Ω, we have

$$ \mathbb{P}\bigl(F(X) - \mathbb{E}F(X) \geq\lambda\sigma\bigr) \leq \exp\bigl(O\bigl(t^2\sigma^2\bigr)-t\lambda\sigma\bigr). $$

Optimizing in t gives the desired result. □