Littlewood-Offord problems for Ising models

TL;DR

This work studies the one-dimensional Littlewood–Offord problem for general Ising spins, establishing universal bounds on the concentration probability $Q_n$ and linking probabilistic concentration to spectral properties. By employing the Edwards–Sokal coupling, the authors reduce the problem to a Rademacher-sum framework over cluster spins, enabling both a general lower bound and a sharp $n^{-1/2}$ upper bound under a finite bound $K$ on couplings and external fields. A key contribution is showing that the number of isolated inner clusters drives the upper bound, with percolation-type estimates ensuring linear growth in $n$, which yields $Q_n \le C(K)n^{-1/2}$. As an application, they derive a spectral lower bound for the truncated covariance matrix of Ising spins, linking concentration phenomena to eigenvalue bounds and providing insight into the dependence structure of spins in Ising models.

Abstract

We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration function \[Q_n(x,v)=P\left(\sum_{i=1}^{n}\varepsilon_iv_i\in(x-1,x+1)\right),\] where $x\in\mathbb{R}$, $v_1,v_2,\ldots,v_n$ are real numbers such that $|v_1|\geq 1, |v_2|\geq 1,\ldots, |v_n|\geq 1$, and $(\varepsilon_i)_{i=1,2,\ldots,n}\in\{-1,1\}^{n}$ are random spins of some Ising model. Let $Q_n=\sup_{x,v}Q_n(x,v)$. Under natural assumptions, we show that there exists a universal constant $C$ such that for all $n\geq 1$, \[\binom{n}{[n/2]}2^{-n}\leq Q_n\leq Cn^{-\frac{1}{2}}.\] As an application of the method, under the same assumption, we give a lower bound on the smallest eigenvalue of the truncated correlation matrix of the Ising model.

Theorems & Definitions (13)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 3.1
• proof
• Lemma 4.1
• Lemma 4.2
• proof : Proof of Lemma \ref{['lem: lower bounds on the number of clusters']}
• Claim 4.3
• Proposition 5.1