Let ε1,…,εn be independent and identically distributed Rademacher random variables taking values ±1 with probability 1/2 each. Given an integer vector a=(a1,…,an), its concentration probability is the quantity ρfalse(bold-italicafalse):=trueprefixsupx∈ZPrfalse(ε1a1+⋯+εnan=xfalse). The Littlewood–Offord problem asks for bounds on ρ(a) under various hypotheses on bold-italica, whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors bold-italica for which ρ(a) is large. In this paper, we study the associated counting problem: How many integer vectors bold-italica belonging to a specified set have large ρ(a)? The motivation for our study is that in typical applications, the inverse Littlewood–Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood–Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first ‘exponential‐type’ (that is, exp(−cnc) for some positive constant c) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best‐known bound is O(n−1/4), due to Cook; and (ii) dense row‐regular {0,1}‐matrices, for which the previous best‐known bound is OCfalse(n−Cfalse) for any constant C>0, due to Nguyen.

Let M n denote a random symmetric n × n matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values 1 and −1 with probability 1/2 each). It is widely conjectured that M n is singular with probability at most (2 + o(1)) −n . On the other hand, the best known upper bound on the singularity probability of M n , due to Vershynin (2011), is 2 −n c , for some unspecified small constant c > 0. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of M n is at most 2 −n 1/4 √ log n/1000 for all sufficiently large n. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij. c n ≥ (1 + o(1))n 2 2 1−n .It has been widely conjectured that this bound is, in fact, tight. On the other hand, perhaps surprisingly, it is non-trivial even to show that c n tends to 0 as n goes to infinity; this was accomplished in a classical work of Komlós in 1967 [6] which showed that c n = O n −1/2 using the classical Erdős-Littlewood-Offord anti-concentration inequality. Subsequently, a breakthrough result due to Kahn, Komlós, and Szemerédi in 1995 [5] showed that c n = O(0.999 n ).Improving upon an intermediate result by Tao and Vu [13], the current 'world record' is c n ≤ (2 + o(1)) −n/2 , due to Bourgain, Vu, and Wood [1].Another widely studied model of random matrices is that of random symmetric matrices; apart from being important for applications, it is also very interesting from a technical perspective as it is one of the simplest models with nontrivial correlations between its entries. Formally, let M n denote a random n × n symmetric matrix, whose upper-diagonal entries are i.i.d. Bernoulli random variables which take values ±1 with probability 1/2 each, and let q n denote the probability that M n is singular. Despite its similarity to c n , much less is known about q n .The problem of whether q n tends to 0 as n goes to infinity was first posed by Weiss in the early 1990s and only settled in 2005 by Costello, Tao, and Vu [2], who showed that q n = O(n −1/8+o(1) ).In order to do this, they introduced and studied a quadratic variant of the Erdős-Littlewood-Offord inequality. Subsequently, Nguyen [7] developed a quadratic variant of inverse Littlewood-Offord theory to show that q n = O C (n −C )for any C > 0, where the implicit constant in O C (·) depends only on C. This so-called quadratic inverse Littlewood-Offord theorem in [7] builds on previous work of Nguyen and Vu [8], which is itself based on deep Freiman-type theorems in additive combinatorics (see [14] and the references therein). The current best known upper bound on q n is due to Vershynin [15], who used a sophisticated and technical geometric framework pioneered by R...

Let p ∈ (0, 1/2) be fixed, and let Bn(p) be an n × n random matrix with i.i.d. Bernoulli random variables with mean p. We show that for all t ≥ 0,where sn(Bn(p)) denotes the least singular value of Bn(p) and Cp, ǫp > 0 are constants depending only on p.

The free energy is a key quantity of interest in Ising models, but unfortunately, computing it in general is computationally intractable. Two popular (variational) approximation schemes for estimating the free energy of general Ising models (in particular, even in regimes where correlation decay does not hold) are: (i) the mean-field approximation with roots in statistical physics, which estimates the free energy from below, and (ii) hierarchies of convex relaxations with roots in theoretical computer science, which estimate the free energy from above. We show, surprisingly, that the tight regime for both methods to compute the free energy to leading order is identical.More precisely, we show that the mean-field approximation is within O((n J F ) 2/3 ) of the free energy, where J F denotes the Frobenius norm of the interaction matrix of the Ising model. This simultaneously subsumes both the breakthrough work of Basak and Mukherjee, who showed the tight result that the mean-field approximation is within o(n) whenever J F = o( √ n), as well as the work of Jain, Koehler, and Mossel, who gave the previously best known non-asymptotic bound of O((n J F ) 2/3 log 1/3 (n J F )). We give a simple, algorithmic proof of this result using a convex relaxation proposed by Risteski based on the Sherali-Adams hierarchy, automatically giving sub-exponential time approximation schemes for the free energy in this entire regime. Our algorithmic result is tight under Gap-ETH. We furthermore combine our techniques with spin glass theory to prove (in a strong sense) the optimality of correlation rounding, refuting a recent conjecture of Allen, O'Donnell, and Zhou. Finally, we give the tight generalization of all of these results to k-MRFs, capturing as a special case previous work on approximating MAX-k-CSP.

No abstract