arXiv subjects

Kostas Zampetakis

Publications and source records attributed to Kostas Zampetakis.

6 recordsLinked to original sources

arXiv2026

On sampling diluted Spin-Glasses with unbounded interactions

Spin-glasses are natural Gibbs distributions that have been studied in Theoretical CS for many decades. Recently, they have been gaining attention from the community as they emerge naturally in neural computation and learning, network inference, optimisation and other areas. We study the problem of efficiently sampling from spin-glass distributions when the underlying graph is a typical instance of $G(n,d/n)$, i.e., the random graph on $n$ vertices such that each edge appears independently with probability $d/n$, and $d=Θ(1)$. Our focus is on the 2-spin model at inverse temperature $β$. We consider this distribution to be one of the most interesting case of spin-glasses, and one of the most challenging to analyse, since its Gaussian couplings give rise to unbounded interaction. We employ the well-known Glauber dynamics to sample from the aforementioned distribution. We show that for the typical instances of the 2-spin model on $G(n,d/n)$, the mixing time of Glauber dynamics is $O\left(n^{1+Θ(\frac{1}{\sqrt{d}})}\right)$, for any $β\leq \frac{1}{4\sqrt{d}}$. Our results can also be adapted for the case of spin-glass distributions with bounded interactions. In that respect, we obtain rapid mixing of Glauber dynamics for the Viana-Bray model on $G(n,d/n)$ when $β\leq \frac{1}{4\sqrt{d}}$. This improves on the current best bound which is $β<\frac{0.18}{\sqrt{d}}$. We utilise stochastic localisation, and in particular, we build and improve on the scheme introduced in [Liu, Mohanty, Rajaraman and Wu: FOCS 2024]. This is the first time that stochastic localisation is used for diluted spin-glasses, where both degrees and interactions can be unbounded.

cs.DM↗

arXiv2026

Fluctuations of the Ising free energy on Erdős-Rényi graphs

We investigate the ferromagnetic Ising model on the Erdős-Rényi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(β,B)$, where $Z_{\mathbb{G}(n,m)}(β,B)$ is the partition function at inverse temperature $β>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $β>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $Θ(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $β<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.

math.CO↗

arXiv2025

The random $k$-SAT Gibbs uniqueness threshold revisited

We prove that for any $k\geq3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the `replica symmetric solution' predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)].The improvement is significant particularly for small $k$.

cs.DM↗

arXiv2024

On sampling diluted Spin Glasses using Glauber dynamics

Spin-glasses are Gibbs distributions that have been studied in CS for many decades. Recently, they have gained renewed attention as they emerge naturally in learning, inference, optimisation etc. We consider the Edwards-Anderson (EA) spin-glass distribution at inverse temperature $β$ when the underlying graph is an instance of $G(n,d/n)$. This is the random graph on $n$ vertices where each edge appears independently with probability $d/n$ and $d=Θ(1)$. We study the problem of approximate sampling from this distribution using Glauber dynamics. For a range of $β$ that depends on $d$ and for typical instances of the EA model on $G(n,d/n)$, we show that the corresponding Glauber dynamics exhibits mixing time $O(n^{2+\frac{3}{\log^2 d}})$. The range of $β$ for which we obtain our rapid-mixing results correspond to the expected influence being $<1/d$; we conjecture that this is the best possible. Unlike the mean-field spin-glasses, where the problem has been studied before, the diluted case has not. We utilise the well-known path-coupling technique. In the standard Glauber dynamics on $G(n,d/n)$, one has to deal with the so-called effect of high degree vertices. Here, rather than considering degrees, it is more natural to use a different measure on the vertices called aggregate influence. We build on the block-construction approach proposed by [Dyer et al. 2006] to circumvent the problem of high-degree vertices. Specifically, we first establish rapid mixing for an appropriately defined block-dynamics. We design this dynamics such that vertices of large aggregate influence are placed deep inside their blocks. Then, we obtain rapid mixing for the Glauber dynamics utilising a comparison argument.

cs.DM↗

arXiv2023

Broadcasting with Random Matrices

Motivated by the theory of spin-glasses in physics, we study the so-called reconstruction problem for the related distributions on the tree, and on the sparse random graph $G(n,d/n)$. Both cases, reduce naturally to studying broadcasting models on the tree, where each edge has its own broadcasting matrix, and this matrix is drawn independently from a predefined distribution. In this context, we study the effect of the configuration at the root to that of the vertices at distance $h$, as $h\to\infty$. We establish the reconstruction threshold for the cases where the broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This threshold seems to be a natural extension of the well-known Kesten-Stigum bound which arises in the classic version of the reconstruction problem. Our results imply, as a special case, the reconstruction threshold for the well-known Edward-Anderson model of spin-glasses on the tree. Also, we extend our analysis to the setting of the Galton-Watson tree, and the random graph $G(n,d/n)$, where we establish the corresponding thresholds.Interestingly, for the Edward-Anderson model on the random graph, we show that the replica symmetry breaking phase transition, established in [Guerra and Toninelli:2004], coincides with the reconstruction threshold. Compared to the classical Gibbs distributions, the spin-glasses have a lot of unique features. In that respect, their study calls for new ideas, e.g., we introduce novel estimators for the reconstruction problem. Furthermore, note that the main technical challenge in the analysis is the presence of (too) many levels of randomness. We manage to circumvent this problem by utilising recently proposed tools coming from the analysis of Markov chains.

cs.DM↗

arXiv2021

The Lovász Local Lemma is Not About Probability

Given a collection of independent events each of which has strictly positive probability, the probability that all of them occur is also strictly positive. The Lovász local lemma (LLL) asserts that this remains true if the events are not too strongly negatively correlated. The formulation of the lemma involves a graph with one vertex per event, with edges indicating potential negative dependence. The word "Local" in LLL reflects that the condition for the negative correlation can be expressed solely in terms of the neighborhood of each vertex. In contrast to this local view, Shearer developed an exact criterion for the avoidance probability to be strictly positive, but it involves summing over all independent sets of the graph. In this work we make two contributions. The first is to develop a hierarchy of increasingly powerful, increasingly non-local lemmata for bounding the avoidance probability from below, each lemma associated with a different set of walks in the graph. Already, at its second level, our hierarchy is stronger than all known local lemmata. To demonstrate its power we prove new bounds for the negative-fugacity singularity of the hard-core model on several lattices, a central problem in statistical physics. Our second contribution is to prove that Shearer's connection between the probabilistic setting and the independent set polynomial holds for \emph{arbitrary supermodular} functions, not just probability measures. This means that all LLL machinery can be employed to bound from below an arbitrary supermodular function, based only on information regarding its value at singleton sets and partial information regarding their interactions. We show that this readily implies both the quantum LLL of Ambainis, Kempe, and Sattath~[JACM 2012], and the quantum Shearer criterion of Sattath, Morampudi, Laumann, and Moessner~[PNAS 2016].

math.PR↗