Applications of Stein's method for concentration inequalities

TL;DR

This paper extends Stein's method to derive new concentration inequalities and applies them to models like Curie--Weiss, Erdős–Rényi graphs, and Ising models, providing insights into their probabilistic behaviors.

Contribution

It introduces theoretical extensions of Stein's method and demonstrates their application to complex dependent systems, yielding new concentration and large deviation results.

Findings

Concentration inequality for magnetization at critical temperature in Curie--Weiss model.

Exact large deviation asymptotics for triangle counts in Erdős–Rényi graphs.

New concentration inequalities for Ising models valid at all temperatures.

Abstract

Stein's method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie--Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erd\H{o}s--R\'{e}nyi random graph $G(n,p)$ when $p\ge0.31$. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.