We study the complexity of approximating the partition function of dense Ising models in the critical regime. Recent work of Chen, Chen, Yin, and Zhang (FOCS 2025) established fast mixing at criticality, and even beyond criticality in a window of width N^-1/2. We complement these algorithmic results by proving nearly tight hardness bounds, thus yielding the first instance of a sharp scaling window for the computational complexity of approximate counting. Specifically, for the dense Ising model we show that approximating the partition function is computationally hard within a window of width N^-1/2+ε for any constant ε>0. Standard hardness reductions for non-critical regimes break down at criticality due to bigger fluctuations in the underlying gadgets, leading to suboptimal bounds. We overcome this barrier via a global approach which aggregates fluctuations across all gadgets rather than requiring tight concentration guarantees for each individually. This new approach yields the optimal exponent for the critical window.

A set in R-d is star-shaped if there is a point in the set that can see every other point in the set in the sense that the line-segment connecting the points lies within the set. We show that testing whether a non-empty compact smooth region is star-shaped is for all R-complete. Since the obvious definition of star-shapedness has logical form there exists for all, this is a somewhat surprising result, based on Krasnosel'skii's theorem from convex geometry; we study several related complexity classifications in the real hierarchy based on other results from convex geometry.

A set is star-shaped if there is a point in the set that can see every other point in the set in the sense that the line-segment connecting the points lies within the set. We show that testing whether a non-empty compact smooth region is star-shaped is $\forall\mathbb{R}$-complete. Since the obvious definition of star-shapedness has logical form $\exists\forall$, this is a somewhat surprising result, based on Krasnosel'skiĭ's theorem from convex geometry; we study several related complexity classifications in the real hierarchy based on other results from convex geometry.

We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution \(\mu\) , an \(\varepsilon \gt 0\) , and access to sampling oracle(s) for a hidden distribution \(\pi\) , the goal in identity testing is to distinguish whether the two distributions \(\mu\) and \(\pi\) are identical or are at least \(\varepsilon\) -far apart. When there is only access to full samples from the hidden distribution \(\pi\) , it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various “conditional” sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the \(\mathsf{Coordinate\ Oracle}\) , and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an \(n\) -dimensional visible distribution \(\mu\) , then there is an efficient identity testing algorithm for any hidden distribution \(\pi\) using \(\widetilde{O}(n/\varepsilon)\) queries to the \(\mathsf{Coordinate\ Oracle}\) . Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of \(n\) -dimensional distributions, specifically sparse antiferromagnetic Ising models over \(\{+1,-1\}^{n}\) , we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless \(\mathsf{RP}=\mathsf{NP}\) . We complement our results with a matching \(\Omega(n/\varepsilon)\) statistical lower bound for the sample complexity of identity testing in the \(\mathsf{Coordinate\ Oracle}\) model.

We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter $\lambda>0$; the special case $\lambda=1$ corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete $\Delta$-regular tree for all $\lambda$. However, Restrepo et al. (2014) showed that for sufficiently large $\lambda$ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width. We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of $O(n)$ for the Glauber dynamics for unweighted independent sets on arbitrary trees. We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree $\Delta$. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order $\lambda=O(1/\Delta)$. Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance via a non-trivial inductive proof.

These are self-contained lecture notes for spectral independence. For an $n$-vertex graph, the spectral independence condition is a bound on the maximum eigenvalue of the $n\times n$ influence matrix whose entries capture the influence between pairs of vertices, it is closely related to the covariance matrix. We will present recent results showing that spectral independence implies the mixing time of the Glauber dynamics is polynomial (where the degree of the polynomial depends on certain parameters). The proof utilizes local-to-global theorems which we will detail in these notes. Finally, we will present more recent results showing that spectral independence implies an optimal bound on the relaxation time (inverse spectral gap) and with some additional conditions implies an optimal mixing time bound of $O(n\log{n})$ for the Glauber dynamics. We also present the results of Anari, Liu, Oveis Gharan, and Vinzant (2019) for generating a random basis of a matroid. The analysis of the associated bases-exchange walk utilizes the local-to-global theorems used for spectral independence with the Trickle-Down Theorem of Oppenheim (2018) to analyze the local walks. Our focus in these notes is on the analysis of the spectral gap of the associated Markov chains from a functional analysis perspective, and we present proofs of the associated local-to-global theorems from this same Markov chain perspective.

The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. (2003), imply Ω(1) spectral gap and O(log n) mixing time, respectively, for the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish Θ(log n) mixing for the Swendsen-Wang dynamics for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model, and for the q-state Potts model when q is small with respect to d. Our proofs feature a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders and utilize recent work on block factorization of entropy under spatial mixing conditions.

We study the performance of Markov chains for the q -state ferromagnetic Potts model on random regular graphs. While the cases of the grid and the complete graph are by now well-understood, the case of random regular graphs has resisted a detailed analysis and, in fact, even analysing the properties of the Potts distribution has remained elusive. It is conjectured that the performance of Markov chains is dictated by metastability phenomena, i.e., the presence of “phases” (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape, and therefore cause slow mixing. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these phases based on optimisation arguments fall short of the task. Our first contribution is to detail the emergence of the two relevant phases for the q -state Potts model on the d -regular random graph for all integers q,d≥ 3 , and establish that for an interval of temperatures, delineated by the uniqueness and a broadcasting threshold on the d -regular tree, the two phases coexist (as possible metastable states). The proofs are based on a conceptual connection between spatial properties and the structure of the Potts distribution on the random regular graph, rather than complicated moment calculations. This significantly refines earlier results by Helmuth, Jenssen, and Perkins who had established phase coexistence for a small interval around the so-called ordered-disordered threshold (via different arguments) that applied for large q and d≥ 5 . Based on our new structural understanding of the model, our second contribution is to obtain metastability results for two classical Markov chains for the Potts model. We first complement recent fast mixing results for Glauber dynamics by Blanca and Gheissari below the uniqueness threshold, by showing an exponential lower bound on the mixing time above the uniqueness threshold. Then, we obtain tight results even for the non-local and more elaborate Swendsen–Wang chain, where we establish slow mixing/metastability for the whole interval of temperatures where the chain is conjectured to mix slowly on the random regular graph. The key is to bound the conductance of the chains using a random graph “planting” argument combined with delicate bounds on random-graph percolation.

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ≥2, the Shearer threshold is the value λ⁎(Δ)=(Δ−1)Δ−1/ΔΔ. It is known that for λ<−λ⁎(Δ), there are graphs G with maximum degree Δ whose independent set polynomial, evaluated at λ, is at most 0. Also, there are no such graphs for any λ>−λ⁎(Δ). This paper is motivated by the computational problem of approximating the independent set polynomial when λ<−λ⁎(Δ). The key issue in complexity bounds for this problem is “implementation”. Informally, an implementation of a real number λ′ is a graph whose hard-core partition function, evaluated at λ, simulates a vertex-weight of λ′ in the sense that λ′ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any λ<−λ⁎(Δ), it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any λ>λ⁎(Δ). Our result has already been used in a paper with Bezáková (STOC 2018) to show that it is #P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most Δ at any value λ<−λ⁎(Δ). In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness).

We show that for every n there are two simple curves on the torus intersecting at least n times without the two curves folding or spiraling with respect to each other. On the other hand, two simple curves in a punctured plane that intersect at least n times (and do not create any empty bigons) must either form a spiral of depth d or a fold of width cn/(d+1)−1, where c only depends on the number of punctures in the plane. The construction of the two curves on the torus involves train tracks and word equations, and the verification that the two curves do not spiral leads us to an infinite binary word based on the golden ratio which does not contain any square word ww for which |w| is even.

We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.

For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the convergence to stationarity and analytic tools for analyzing the decay of relative entropy. As a corollary of our general results, we obtain $O(n\log{n})$ mixing time and $\Omega(1/n)$ modified log-Sobolev constant of the Glauber dynamics for sampling random $q$-colorings of an $n$-vertex graph with constant maximum degree $\Delta$ when $q > (11/6 - \epsilon_0)\Delta$ for some fixed $\epsilon_0>0$. We also obtain $O(\log{n})$ mixing time and $\Omega(1)$ modified log-Sobolev constant of the Swendsen-Wang dynamics for the ferromagnetic Ising model on an $n$-vertex graph of constant maximum degree when the parameters of the system lie in the tree uniqueness region. At the heart of our results are new techniques for establishing spectral independence of the spin system and block factorization of the relative entropy. On one hand we prove that a contractive coupling of a local Markov chain implies spectral independence of the Gibbs distribution. On the other hand we show that spectral independence implies factorization of entropy for arbitrary blocks, establishing optimal bounds on the modified log-Sobolev constant of the corresponding block dynamics.

We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. Here, we instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.

We present improved algorithms and matching statistical and computational lower bounds for the problem of identity testing n-dimensional distributions. In the identity testing problem, we are given as input an explicit distribution μ, an ε > 0, and access to a sampling oracle for a hidden distribution π. The goal is to distinguish whether the two distributions μ and π are identical or are at least ε-far apart. When there is only access to full samples from the hidden distribution π, it is known that exponentially many samples may be needed for identity testing, and hence previous works have studied identity testing with additional access to various conditional sampling oracles. We consider here a significantly weaker conditional sampling oracle, which we call the Coordinate Oracle, and provide a fairly complete computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for the visible distribution μ, then there is an efficient identity testing algorithm for any hidden π that uses Õ(n/ε) queries to the Coordinate Oracle. Approximate tensorization of entropy is a classical tool for proving optimal mixing time bounds of Markov chains and concentration of Lipschitz functions for high-dimensional distributions. Recent work shows that spectral independence implies approximate tensorization and consequently establishes it for many families of n-dimensional distributions. We complement our algorithmic result for identity testing with a matching Ω(n/ε) statistical lower bound for the number of queries under the Coordinate Oracle. We also prove a computational phase transition: for a well-studied class of n-dimensional distributions, specifically sparse antiferromagnetic Ising models over {+1,−1}n, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless RP=NP. Department of Computer Science and Engineering, Pennsylvania State University. Email: ablanca@cse.psu.edu. Research supported in part by NSF grant CCF-1850443. Department of Mathematics, Massachusetts Institute of Technology. Email: zongchen@mit.edu. Department of Computer Science, University of Rochester. Email: stefanko@cs.rochester.edu. Research supported in part by NSF grant CCF-1563757. Department of Computer Science, University of California, Santa Barbara. Email: vigoda@ucsb.edu. Research supported in part by NSF grant CCF-2147094.

Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n) , where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.

For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bounded-degree graphs. We present new algorithmic results for approximating the partition function and sampling from the Gibbs distribution for spin systems in the non-uniqueness region on random regular bipartite graphs. We give an $\mathsf{FPRAS}$ for counting $q$-colorings for even $q=O\big(\tfrac{\Delta}{\log{\Delta}}\big)$ on almost every $\Delta$-regular bipartite graph. This is within a factor $O(\log{\Delta})$ of the sampling algorithm for general graphs in the uniqueness region and improves significantly upon the previous best bound of $q=O\big(\tfrac{\sqrt{\Delta}}{(\log\Delta)^2}\big)$ by Jenssen, Keevash, and Perkins (SODA'19). Analogously, for the hard-core model on independent sets weighted by $\lambda>0$, we present an $\mathsf{FPRAS}$ for estimating the partition function when $\lambda=\Omega\big(\tfrac{\log{\Delta}}{\Delta}\big)$, which improves upon previous results by an $\Omega(\log \Delta)$ factor. Our results for the colorings and hard-core models follow from a general result that applies to arbitrary spin systems. Our main contribution is to show how to elevate probabilistic/analytic bounds on the marginal probabilities for the typical structure of phases on random bipartite regular graphs into efficient algorithms, using the polymer method. We further show evidence that our result for colorings is within a constant factor of best possible using current polymer-method approaches.

for each i ∈ [m] and j ∈ [n] (we use the notation [k] = {1, . . . , k}). Notice that it must be that ∑ ri = ∑ cj for such a contingency table to possibly exist. We are motivated by the problem of sampling uniformly from Ω(r, c), the space of all contingency tables with row sums r and column sums c. Throughout the majority of this discussion, the primary method of sampling will be using Markov chains. The definitions and main results dealing with Markov chains are primarily acquired from Mark Jerrum’s textbook Counting, Sampling and Integrating: Algorithms and Complexity [9]. A (finitespace) Markov chain is a sequence of random variables X0, . . . , Xt, . . . with values in a finite state space Ω such that

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems. Let $\alpha^*\approx 1.763$ denote the solution to $\exp(1/x)=x$ and let $\alpha>\alpha^*$. We prove that, for any triangle-free graph $G=(V,E)$ with maximum degree $\Delta$, for all $q\geq\alpha\Delta+1$, the mixing time of the Glauber dynamics for $q$-colorings is polynomial in $n=|V|$, with the exponent of the polynomial independent of $\Delta$ and $q$. In comparison, previous approximate counting results for colorings held for a similar range of $q$ (asymptotically in $\Delta$) but with larger girth requirement or with a running time where the polynomial exponent depended on $\Delta$ and $q$ (exponentially). One further feature of using the spectral independence approach to study colorings is that it avoids many of the technical complications in previous approaches caused by coupling arguments or by passing to the complex plane; the key improvement on the running time is based on relatively simple combinatorial arguments which are then translated into spectral bounds.