Alex Zhai

Exponential concentration of cover times

We prove an exponential concentration bound for cover times of general graphs in terms of the Gaussian free field, extending the work of Ding-Lee-Peres and Ding. The estimate is asymptotically sharp as the ratio of hitting time to cover time goes to zero. The bounds are obtained by showing a stochastic domination in the generalized second Ray-Knight theorem, which was shown to imply exponential concentration of cover times by Ding. This stochastic domination result appeared earlier in a preprint of Lupu, but the connection to cover times was not mentioned.

On multiple peaks and moderate deviations for the supremum of a Gaussian field

We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns with the concept of multiple peaks. A theorem of Chatterjee states that when a centered Gaussian field admits the so-called superconcentration property, it typically attains values near its maximum on multiple near-orthogonal sites, known as multiple peaks. We improve his theorem in two aspects: (i) the number of peaks attained by our bound is of the order

Gravitational allocation on the sphere

\exp(c / \sigma^2)

Average-Case Reconstruction for the Deletion Channel: Subpolynomially Many Traces Suffice

(as opposed to Chatterjees polynomial bound in

Fibonacci-like growth of numerical semigroups of a given genus

1/\sigma

Surprise probabilities in Markov chains

), where

Gravitational allocation for uniform points on the sphere

\sigma

When multiplicative noise stymies control.

is the standard deviation of the supremum of the Gaussian field, which is assumed to have variance at most

The CLT in high dimensions: quantitative bounds via martingale embedding.

1

A high-dimensional CLT in

and (ii) our bound need not assume that the correlations are non-negative. We also prove a similar result based on the superconcentration of the free energy. As primary applications, we infer that for the S-K spin glass model on the