arXiv subjects

Will Perkins

Publications and source records attributed to Will Perkins.

At least 55 records · Page 3Linked to original sources

arXiv2022

Independent sets in the hypercube revisited

We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets in the discrete hypercube $\{0,1\}^d$ and Galvin's follow-up work on weighted independent sets. We combine Sapozhenko's graph container methods with the cluster expansion and abstract polymer models, two tools from statistical physics, to obtain considerably sharper asymptotics and detailed probabilistic information about the typical structure of (weighted) independent sets in the hypercube. These results refine those of Korshunov and Sapozhenko and Galvin, and answer several questions of Galvin.

math.CO↗

arXiv2021

Three simple scenarios for high-dimensional sphere packings

Based on results from the physics and mathematics literature which suggest a series of clearly defined conjectures, we formulate three simple scenarios for the fate of hard sphere crystallization in high dimension: (A) crystallization is impeded and the glass phase constitutes the densest packing, (B) crystallization from the liquid is possible, but takes place much beyond the dynamical glass transition and is thus dynamically implausible, or (C) crystallization is possible and takes place before (or just after) dynamical arrest, thus making it plausibly accessible from the liquid state. In order to assess the underlying conjectures and thus obtain insight into which scenario is most likely to be realized, we investigate the densest sphere packings in dimension $d=3$-$10$ using cell-cluster expansions as well as numerical simulations. These resulting estimates of the crystal entropy near close-packing tend to support scenario C. We additionally confirm that the crystal equation of state is dominated by the free volume expansion and that a meaningful polynomial correction can be formulated.

cond-mat.stat-mech↗

arXiv2021

Computational thresholds for the fixed-magnetization Ising model

The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair. Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins). In particular, letting $β_c(Δ)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $Δ$-regular tree, and $η_{Δ,β,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $Δ$-regular tree at inverse temperature $β$, we prove, for the class of graphs of maximum degree $Δ$: 1. For $β< β_c(Δ)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $η$. 2. For $β> β_c(Δ)$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| >η_{Δ,β,1}^+ $. 3. For $β> β_c(Δ)$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| <η_{Δ,β,1}^+ $ unless NP=RP\@.

cs.DS↗

arXiv2021

Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

For $Δ\ge 5$ and $q$ large as a function of $Δ$, we give a detailed picture of the phase transition of the random cluster model on random $Δ$-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random $Δ$-regular graphs at all temperatures when $q$ is large. This includes the critical temperature at which it is known the Glauber and Swendsen-Wang dynamics for the Potts model mix slowly. We further prove new slow-mixing results for Markov chains, most notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an open interval containing the critical temperature. This was previously only known at the critical temperature. Many of our results apply more generally to $Δ$-regular graphs satisfying a small-set expansion condition.

math.PR↗

arXiv2021

Approximately counting independent sets in bipartite graphs via graph containers

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs satisfying a weak expansion condition: when $d$ is constant, and the graph is a bipartite $Ω( \log^2 d/d)$-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for $d>5$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a $d$-regular, bipartite $α$-expander, with $α>0$ fixed, we give an FPTAS for the hard-core model partition function at fugacity $λ=Ω(\log d / d^{1/4})$. Finally we present an algorithm that applies to all $d$-regular, bipartite graphs, runs in time $\exp\left( O\left( n \cdot \frac{ \log^3 d }{d } \right) \right)$, and outputs a $(1 + o(1))$-approximation to the number of independent sets.

cs.DS↗

arXiv2021

Potential-weighted connective constants and uniqueness of Gibbs measures

We define a potential-weighted connective constant that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on $\mathbb R^d$ and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the potential-weighted connective constant this object exhibits an infinite volume uniqueness property. On the other hand, we show that our uniqueness bound can be tight for certain spaces: the same infinite volume object exhibits non-uniqueness for activities above our bound in the case when the underlying space has the geometry of a tree.

math.PR↗

arXiv2021

Approximation algorithms for the random-field Ising model

Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $Δ$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $Δ$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $Θ(\log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.

cs.DS↗

arXiv2021

Approximate counting and sampling via local central limit theorems

We give an FPTAS for computing the number of matchings of size $k$ in a graph $G$ of maximum degree $Δ$ on $n$ vertices, for all $k \le (1-δ)m^*(G)$, where $δ>0$ is fixed and $m^*(G)$ is the matching number of $G$, and an FPTAS for the number of independent sets of size $k \le (1-δ) α_c(Δ) n$, where $α_c(Δ)$ is the NP-hardness threshold for this problem. We also provide quasi-linear time randomized algorithms to approximately sample from the uniform distribution on matchings of size $k \leq (1-δ)m^*(G)$ and independent sets of size $k \leq (1-δ)α_c(Δ)n$. Our results are based on a new framework for exploiting local central limit theorems as an algorithmic tool. We use a combination of Fourier inversion, probabilistic estimates, and the deterministic approximation of partition functions at complex activities to extract approximations of the coefficients of the partition function. For our results for independent sets, we prove a new local central limit theorem for the hard-core model that applies to all fugacities below $λ_c(Δ)$, the uniqueness threshold on the infinite $Δ$-regular tree.

cs.DS↗

arXiv2021

A proof of the Upper Matching Conjecture for large graphs

We prove that the `Upper Matching Conjecture' of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every $d$ and every large enough $n$ divisible by $2d$, a union of $n/(2d)$ copies of the complete $d$-regular bipartite graph maximizes the number of independent sets and matchings of size $k$ for each $k$ over all $d$-regular graphs on $n$ vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.

math.CO↗

arXiv2021

Correlation decay for hard spheres via Markov chains

We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of $2$ and $1.7$, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results.

math.PR↗

arXiv2021

On the Number of Independent Sets in Uniform, Regular, Linear Hypergraphs

We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) $r\ge 3$, with $d$ and $n$ going to infinity. In the case of strong independent sets, for $r=3$, we provide an upper bound that is tight up to the second-order term, improving on a result of Ordentlich-Roth (2004). The tightness in the strong independent set case is established by an explicit construction of a $3$-uniform, $d$-regular, cross-edge free, linear hypergraph on $n$ vertices which could be of interest in other contexts. We leave open the general case(s) with some conjectures. Our proofs use the occupancy method introduced by Davies, Jenssen, Perkins, and Roberts (2017).

math.CO↗

arXiv2021

Fast algorithms at low temperatures via Markov chains

We define a discrete-time Markov chain for abstract polymer models and show that under sufficient decay of the polymer weights, this chain mixes rapidly. We apply this Markov chain to polymer models derived from the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs. In this setting, Jenssen, Keevash and Perkins (2019) recently gave an FPTAS and an efficient sampling algorithm at sufficiently high fugacity and low temperature respectively. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. Our approach via the polymer model Markov chain circumvents the zero-free analysis and the generalization to complex parameters, and leads to a sampling algorithm with a fast running time of $O(n \log n)$ for the Potts model and $O(n^2 \log n)$ for the hard-core model, in contrast to typical running times of $n^{O(\log Δ)}$ for algorithms based on Barvinok's polynomial interpolation method on graphs of maximum degree $Δ$. We finally combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin Glauber dynamics restricted to even and odd or `red' dominant portions of the respective state spaces.

cs.DS↗

arXiv2021

Frozen $1$-RSB structure of the symmetric Ising perceptron

We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB' structure conjectured by Krauth and Mezard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.

math.PR↗

arXiv2020

Maximum entropy and integer partitions

We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions.

math.CO↗

arXiv2020

Algorithms for #BIS-hard problems on expander graphs

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) $Δ$-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite $Δ$-regular tree. We also find efficient counting and sampling algorithms for proper $q$-colorings of random $Δ$-regular bipartite graphs when $q$ is sufficiently small as a function of $Δ$.

cs.DS↗

arXiv2019

Independent Sets, Matchings, and Occupancy Fractions

We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and the independence polynomial of a d-regular graph. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of $K_{d,d}$. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström. In probabilistic language, our main theorems state that for all d-regular graphs and all $λ$, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity $λ$ are maximized by $K_{d,d}$. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case.

math.CO↗

arXiv2019

Counting independent sets in unbalanced bipartite graphs

We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $λ=1$ (approximating the number of independent sets of $G$) and $Δ_R \geq 7Δ_L \log(Δ_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.

cs.DS↗

arXiv2019

Storage capacity in symmetric binary perceptrons

We study the problem of determining the capacity of the binary perceptron for two variants of the problem where the corresponding constraint is symmetric. We call these variants the rectangle-binary-perceptron (RPB) and the $u-$function-binary-perceptron (UBP). We show that, unlike for the usual step-function-binary-perceptron, the critical capacity in these symmetric cases is given by the annealed computation in a large region of parameter space (for all rectangular constraints and for narrow enough $u-$function constraints, $K K^*$. We conclude that full-step-replica-symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case.

cond-mat.dis-nn↗