Universality of the mean-field for the Potts model

Basak, Anirban; Mukherjee, Sumit

doi:10.1007/s00440-016-0718-0

Cited by 55 publications

(86 citation statements)

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“…This as a direct consequence reduced the large deviations for subgraph counts in G n,p to a 'meanfield' entropic variational problem in certain regimes of the sparsity parameter p. Thereafter, Eldan [18] obtained an improved set of conditions under which the above reduction holds, approximating the Gibbs measure, up to lower order entropy, by a product measure. Similar results for Gibbs measures beyond the hypercube were obtained by [6,30], and recently by Austin [5] for very general product spaces. Furthermore, using different approaches, the large deviation behavior for several spectral and geometric functionals under almost optimal sparsity assumptions were established independently and simultaneously by Cook and Dembo [16] and Augeri [4].…”

Section: Introductionsupporting

confidence: 84%

“…The independence polynomial of a graph H is defined to be PH (x) := k iH (k)x k , where iH (k) is the number of k-element independent sets in H 6. By the definition of the independence polynomial, for any graph H and vertex v in it,PH (x) = PH 1 (x) + xPH 2 (x),where H1 is obtained from H by deleting v and H2 is obtained from H by deleting v and all its neighbors.…”

mentioning

confidence: 99%

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Upper Tails for Edge Eigenvalues of Random Graphs

Bhattacharya¹,

Ganguly²

2020

SIAM J. Discrete Math.

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In this note we prove a precise large deviation principle for the largest and second largest eigenvalues of a sparse Erdős-Rényi graph. Our arguments rely on various recent breakthroughs in the study of mean field approximations for large deviations of low complexity non-linear functions of independent Bernoulli variables and solutions of the associated entropic variational problems.1 1 Note that A(Gn,p) − p11 ′ is the centered adjacency matrix up to a translation by pI, where I is the identity matrix. Since this deterministically shifts every eigenvalue by p which is asymptotically negligible, we will continue this abuse of terminology.2 We write f g to denote f = O(g), f ∼ g means f = (1 + o(1))g, and f ≪ g means f = o(g).

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Section: Introductionsupporting

confidence: 84%

mentioning

confidence: 99%

Upper Tails for Edge Eigenvalues of Random Graphs

Bhattacharya¹,

Ganguly²

2020

SIAM J. Discrete Math.

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“…We note that [Jain et al, 2018a] prove this inequality is tight up to constants. This also recovers the result of [Basak and Mukherjee, 2017] which shows the error is o(n) when J 2 F = o(n). The full results are in Section 4.…”

Section: Introductionsupporting

confidence: 87%

“…In recent years, considerable effort has gone into bounding the error of the mean-field approximation on more general graphs; we refer the reader to [Basak andMukherjee, 2017, Jain et al, 2018a] for a detailed discussion and comparison of results in this direction. If one only wishes to show that the mean-field approximation asymptotically gives the correct free energy density F /n and does not care about the rate of convergence, then the breakthrough result is due to [Basak and Mukherjee, 2017], who provided an exponential improvement over previous work of [Borgs et al, 2012] to identify the regime where this happens.…”

Section: Introductionmentioning

confidence: 99%

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Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective

Jain

Koehler

Risteski

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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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.

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Limit theorems for monochromatic stars

Bhattacharya

Mukherjee

2019

Random Struct Algorithms

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Let T(K 1,r , G n ) be the number of monochromatic copies of the r-star K 1,r in a uniformly random coloring of the vertices of the graph G n . In this paper we provide a complete characterization of the limiting distribution of T(K 1,r , G n ), in the regime where E(T(K 1,r , G n )) is bounded, for any growing sequence of graphs G n . The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most r. Conversely, any limiting distribution of T(K 1,r , G n ) has a representation of this form. Examples and connections to the birthday problem are discussed. KEYWORDScombinatorial probability, graph coloring, limit theorems, subgraph counts 1 a uv (G n 2) the number of monochromatic edges in G n . Random Struct Alg. 2019;55:831-853. wileyonlinelibrary.com/journal/rsa

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Universality of the mean-field for the Potts model

Cited by 55 publications

References 50 publications

Upper Tails for Edge Eigenvalues of Random Graphs

Upper Tails for Edge Eigenvalues of Random Graphs

Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective

Limit theorems for monochromatic stars

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