arXiv subjects

Will Perkins

Publications and source records attributed to Will Perkins.

At least 37 records · Page 2Linked to original sources

arXiv2025

The typical structure of dense claw-free graphs

We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $γ\in (0,1)$. We show that the problem exhibits a second-order phase transition at edge density $γ^\ast=\frac{5-\sqrt{5}}{4}$. The asymptotic formula arises by solving a variational problem over graphons. For $γ\geqγ^\ast$ there is a unique optimal graphon, while for $γ<γ^\ast$ there is an infinite set of optimal graphons. By analyzing more detailed structure, we prove that for $γ<γ^\ast$, there is in fact a unique graphon $W$ such that almost all claw-free graphs at edge density $γ$ are close in cut metric to $W$. We also analyze the probability of claw-freeness in the Erdős-Rényi random graph $G(n,p)$ for constant $p$, obtaining a formula for the large-deviation rate function for claw-freeness. In this case, the problem exhibits a first-order phase transition at $p^\ast=\frac{3-\sqrt{5}}{2}$, separating distinct structural regimes. At the critical point $p^\ast$, the corresponding graphon variational problem has infinitely many solutions, and we again pinpoint a unique optimal graphon that describes the typical structure of $G(n,p^\ast)$ conditioned on being claw-free.

math.CO↗

arXiv2024

Lower tails for triangles inside the critical window

We study the probability that the random graph $G(n,p)$ is triangle-free. When $p =o(n^{-1/2})$ or $p = ω(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and via regularity or hypergraph container methods in the latter case. We prove for the first time an asymptotic formula for the logarithm of this probability when $p = c n^{-1/2}$ for $c$ a sufficiently small constant. More generally, we study lower-tail large deviations for triangles in random graphs: the probability that $G(n,p)$ has at most $η$ times its expected number of triangles, when $p = c n^{-1/2}$ for $c$ and $η\in [0,1)$ constant. Our results apply for all $c$ if $η\ge .4993$ and for $c$ small enough otherwise. For $η$ small (including the case of triangle-freeness), we prove that a phase transition occurs as $c$ varies, in the sense of a non-analyticity of the rate function, while for $η\ge .4993$ we prove that no phase transition occurs. On the other hand for the random graph $G(n,m)$, with $m = b n^{3/2}$, we show that a phase transition occurs in the lower-tail problem for triangles as $b$ varies for \emph{every} $η\in [0,1)$. Our method involves ingredients from algorithms and statistical physics including the cluster expansion and concentration inequalities for contractive Markov chains.

math.PR↗

arXiv2024

Pirogov--Sinai Theory Beyond Lattices

Pirogov--Sinai theory is a well-developed method for understanding the low-temperature phase diagram of statistical mechanics models on lattices. Motivated by physical and algorithmic questions beyond the setting of lattices, we develop a combinatorially flexible version of Pirogov--Sinai theory for the hard-core model of independent sets. Our results illustrate that the main conclusions of Pirogov--Sinai theory can be obtained in significantly greater generality than that of $\mathbb Z^{d}$. The main ingredients in our generalization are combinatorial and involve developing appropriate definitions of contours based on the notion of cycle basis connectivity. This is inspired by works of Timár and Georgakopoulos--Panagiotis.

math.PR↗

arXiv2024

Sampling and counting triangle-free graphs near the critical density

We study the following combinatorial counting and sampling problems: can we efficiently sample from the Erdős-Rényi random graph $G(n,p)$ conditioned on triangle-freeness? Can we efficiently approximate the probability that $G(n,p)$ is triangle-free? These are prototypical instances of forbidden substructure problems ubiquitous in combinatorics. The algorithmic questions are instances of approximate counting and sampling for a hypergraph hard-core model. Estimating the probability that $G(n,p)$ has no triangles is a fundamental question in probabilistic combinatorics and one that has led to the development of many important tools in the field. Through the work of several authors, the asymptotics of the logarithm of this probability are known if $p =o( n^{-1/2})$ or if $p =ω( n^{-1/2})$. The regime $p = Θ(n^{-1/2})$ is more mysterious, as this range witnesses a dramatic change in the the typical structural properties of $G(n,p)$ conditioned on triangle-freeness. As we show, this change in structure has a profound impact on the performance of sampling algorithms. We give two different efficient sampling algorithms for triangle-free graphs (and complementary algorithms to approximate the triangle-freeness large deviation probability), one that is efficient when $p < c/\sqrt{n}$ and one that is efficient when $p > C/\sqrt{n}$ for constants $c, C>0$. The latter algorithm involves a new approach for dealing with large defects in the setting of sampling from low-temperature spin models.

cs.DS↗

arXiv2024

Hardness of sampling for the anti-ferromagnetic Ising model on random graphs

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree $d$ for large constant $d$, proving that when the normalized inverse temperature satisfies $β>1$ (asymptotically corresponding to the condensation threshold), then w.h.p. over the random graph there is no stable sampling algorithm that can output a sample close in $W_2$ distance to the Gibbs measure. The results also apply to a fixed-magnetization version of the model, showing that there are no stable sampling algorithms for low but positive temperature max and min bisection distributions. These results show a gap in the tractability of search and sampling problems: while there are efficient algorithms to find near optimizers, stable sampling algorithms cannot access the Gibbs distribution concentrated on such solutions. Our techniques involve extensions of the interpolation technique relating behavior of the mean field Sherrington-Kirkpatrick model to behavior of Ising models on random graphs of average degree $d$ for large $d$. While previous interpolation arguments compared the free energies of the two models, our argument compares the average energies and average overlaps in the two models.

math.PR↗

arXiv2024

Perfect Sampling for Hard Spheres from Strong Spatial Mixing

We provide a perfect sampling algorithm for the hard-sphere model on subsets of $\mathbb{R}^d$ with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.

cs.DS↗

arXiv2024

Searching for (sharp) thresholds in random structures: where are we now?

We survey the current state of affairs in the study of thresholds and sharp thresholds in random structures on the occasion of the recent proof of the Kahn--Kalai Conjecture by Park and Pham and the fairly recent proof of the satisfiability conjecture for large $k$ by Ding, Sly, and Sun. Random discrete structures appear as fundamental objects of study in many scientific and mathematical fields including statistical physics, combinatorics, algorithms and complexity, social choice theory, coding theory, and statistics. While the models and properties of interest in these fields vary widely, much progress has been made through the development of general tools applicable to large families of models and properties all at once. Historically these tools originated to solve or make progress on specific, difficult conjectures in the areas mentioned above. We will survey recent progress on some of these hard problems and describe some challenges for the future.

math.CO↗

arXiv2023

Percolation on hypergraphs and the hard-core model

We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $Δ$ and for $k$-uniform hypergraphs of maximum degree $Δ$ in which any pair of edges overlaps in at most $r$ vertices. The hypergraphs that achieve these bounds are hypertrees, but unlike in the case of graphs, there are many different $k$-uniform, $Δ$-regular hypertrees. Determining the extremal tree for a given $k, Δ, r$ involves an optimization problem, and our bounds arise from a convex relaxation of this problem. By combining our percolation bounds with the method of disagreement percolation we obtain improved bounds on the uniqueness threshold for the hard-core model on hypergraphs satisfying the same constraints. Our uniqueness conditions imply exponential weak spatial mixing, and go beyond the known bounds for rapid mixing of local Markov chains and existence of efficient approximate counting and sampling algorithms. Our results lead to natural conjectures regarding the aforementioned algorithmic tasks, based on the intuition that uniqueness thresholds for the extremal hypertrees for percolation determine computational thresholds.

math.PR↗

arXiv2023

On the hardness of finding balanced independent sets in random bipartite graphs

We consider the algorithmic problem of finding large \textit{balanced} independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree $d$, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density $(2+o_d(1)) \frac{\log d}{d}$. Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than $(1+ε) \frac{\log d}{d}$ for any $ε>0$ fixed and $d$ large but constant. This factor $2$ statistical--computational gap between what exists and what local algorithms can achieve is analogous to the gap for finding large independent sets in (non-bipartite) random graphs. Our results therefor suggest that this gap is pervasive in many models, and that hard computational problems can lurk inside otherwise tractable ones. A particularly striking aspect of the gap in bipartite graphs is that the algorithm achieving the lower bound is extremely simple and can be implemented as a $1$-local algorithm and a degree-$1$ polynomial (a linear function).

cs.DS↗

arXiv2023

Algorithmic Pirogov-Sinai theory

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $\mathbb Z^d$ and on the torus $(\mathbb Z/n \mathbb Z)^d$. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $\mathbb Z^d$ with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $(\mathbb Z/n \mathbb Z)^d$ at sufficiently low temperature.

cs.DS↗

arXiv2023

Geometric Barriers for Stable and Online Algorithms for Discrepancy Minimization

For many computational problems involving randomness, intricate geometric features of the solution space have been used to rigorously rule out powerful classes of algorithms. This is often accomplished through the lens of the multi Overlap Gap Property ($m$-OGP), a rigorous barrier against algorithms exhibiting input stability. In this paper, we focus on the algorithmic tractability of two models: (i) discrepancy minimization, and (ii) the symmetric binary perceptron (\texttt{SBP}), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. Our first focus is on the limits of online algorithms. By establishing and leveraging a novel geometrical barrier, we obtain sharp hardness guarantees against online algorithms for both the \texttt{SBP} and discrepancy minimization. Our results match the best known algorithmic guarantees, up to constant factors. Our second focus is on efficiently finding a constant discrepancy solution, given a random matrix $\mathcal{M}\in\mathbb{R}^{M\times n}$. In a smooth setting, where the entries of $\mathcal{M}$ are i.i.d. standard normal, we establish the presence of $m$-OGP for $n=Θ(M\log M)$. Consequently, we rule out the class of stable algorithms at this value. These results give the first rigorous evidence towards a conjecture of Altschuler and Niles-Weed~\cite[Conjecture~1]{altschuler2021discrepancy}. Our methods use the intricate geometry of the solution space to prove tight hardness results for online algorithms. The barrier we establish is a novel variant of the $m$-OGP. Furthermore, it regards $m$-tuples of solutions with respect to correlated instances, with growing values of $m$, $m=ω(1)$. Importantly, our results rule out online algorithms succeeding even with an exponentially small probability.

cs.CC↗

arXiv2023

Approximately counting independent sets of a given size in bounded-degree graphs

We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $α_c(Δ)$ and provide (i) for $α< α_c(Δ)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $αn$ in $n$-vertex graphs of maximum degree $Δ$; and (ii) a proof that unless NP=RP, no such algorithms exist for $α>α_c(Δ)$. The critical density is the occupancy fraction of the hard core model on the complete graph $K_{Δ+1}$ at the uniqueness threshold on the infinite $Δ$-regular tree, giving $α_c(Δ)\sim\frac{e}{1+e}\frac{1}Δ$ as $Δ\to\infty$. Our methods apply more generally to anti-ferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.

cs.DS↗

arXiv2022

On the zeroes of hypergraph independence polynomials

We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $Δ$ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree $Δ$ established by Shearer (of radius $\sim 1/(e Δ)$). Up to logarithmic factors in $Δ$ this is optimal, even for hypergraphs with all edge-sizes strictly greater than $2$. We conjecture that for $k\ge 3$, $k$-uniform linear hypergraphs have a much larger zero-free disk of radius $Ω(Δ^{- \frac{1}{k-1}} )$. We establish this in the case of linear hypertrees.

math.CO↗

arXiv2022

Analyticity for classical gasses via recursion

We give a new criterion for a classical gas with a repulsive pair potential to exhibit uniqueness of the infinite volume Gibbs measure and analyticity of the pressure. Our improvement on the bound for analyticity is by a factor $e^2$ over the classical cluster expansion approach and a factor $e$ over the known limit of cluster expansion convergence. The criterion is based on a contractive property of a recursive computation of the density of a point process. The key ingredients in our proofs include an integral identity for the density of a Gibbs point process and an adaptation of the algorithmic correlation decay method from theoretical computer science. We also deduce from our results an improved bound for analyticity of the pressure as a function of the density.

math-ph↗

arXiv2022

Strong spatial mixing for repulsive point processes

We prove that a Gibbs point process interacting via a finite-range, repulsive potential $ϕ$ exhibits a strong spatial mixing property for activities $λ< e/Δ_ϕ$, where $Δ_ϕ$ is the potential-weighted connective constant of $ϕ$, defined recently in [MP21]. Using this we derive several analytic and algorithmic consequences when $λ$ satisfies this bound: (1) We prove new identities for the infinite volume pressure and surface pressure of such a process (and in the case of the surface pressure establish its existence). (2) We prove that local block dynamics for sampling from the model on a box of volume $N$ in $\mathbb R^d$ mixes in time $O(N \log N)$, giving efficient randomized algorithms to approximate the partition function and approximately sample from these models. (3) We use the above identities and algorithms to give efficient approximation algorithms for the pressure and surface pressure.

math.PR↗

arXiv2022

Efficient sampling and counting algorithms for the Potts model on $\mathbb Z^d$ at all temperatures

For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on finite tori $(\mathbb Z / n \mathbb Z )^d$ for any inverse temperature $β\geq 0$. This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen--Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models at all temperatures. Our algorithms are based on representing the random cluster model as a contour model using Pirogov--Sinai theory, and then computing an accurate approximation of the logarithm of the partition function by inductively truncating the resulting cluster expansion. The main innovation of our approach is an algorithmic treatment of unstable ground states, which is essential for our algorithms to apply to all inverse temperatures $β$. By treating unstable ground states our work gives a general template for converting probabilistic applications of Pirogov-Sinai theory to efficient algorithms.

math.PR↗

arXiv2022

Algorithms and Barriers in the Symmetric Binary Perceptron Model

The symmetric binary perceptron ($\texttt{SBP}$) exhibits a dramatic statistical-to-computational gap: the densities at which known efficient algorithms find solutions are far below the threshold for the existence of solutions. Furthermore, the $\texttt{SBP}$ exhibits a striking structural property: at all positive constraint densities almost all of its solutions are 'totally frozen' singletons separated by large Hamming distance \cite{perkins2021frozen,abbe2021proof}. This suggests that finding a solution to the $\texttt{SBP}$ may be computationally intractable. At the same time, the $\texttt{SBP}$ does admit polynomial-time search algorithms at low enough densities. A conjectural explanation for this conundrum was put forth in \cite{baldassi2020clustering}: efficient algorithms succeed in the face of freezing by finding exponentially rare clusters of large size. However, it was discovered recently that such rare large clusters exist at all subcritical densities, even at those well above the limits of known efficient algorithms \cite{abbe2021binary}. Thus the driver of the statistical-to-computational gap exhibited by this model remains a mystery. In this paper, we conduct a different landscape analysis to explain the algorithmic tractability of this problem. We show that at high enough densities the $\texttt{SBP}$ exhibits the multi Overlap Gap Property ($m-$OGP), an intricate geometrical property known to be a rigorous barrier for large classes of algorithms. Our analysis shows that the $m-$OGP threshold (a) is well below the satisfiability threshold; and (b) matches the best known algorithmic threshold up to logarithmic factors as $m\to\infty$. We then prove that the $m-$OGP rules out the class of stable algorithms for the $\texttt{SBP}$ above this threshold. We conjecture that the $m \to \infty$ limit of the $m$-OGP threshold marks the algorithmic threshold for the problem.

cs.CC↗

arXiv2022

Independent sets of a given size and structure in the hypercube

We determine the asymptotics of the number of independent sets of size $\lfloor β2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $β\in [0,1]$ as $d \to \infty$, extending a result of Galvin for $β\in [1-1/\sqrt{2},1]$. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard core model at any fixed fugacity $λ>0$. In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

math.CO↗