# Computational Transition at the Uniqueness Threshold

@article{Sly2010ComputationalTA,
title={Computational Transition at the Uniqueness Threshold},
author={Allan Sly},
journal={2010 IEEE 51st Annual Symposium on Foundations of Computer Science},
year={2010},
pages={287-296}
}
• Allan Sly
• Published 30 May 2010
• Mathematics, Computer Science
• 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
The hardcore model is a model of lattice gas systems which has received much attention in statistical physics, probability theory and theoretical computer science. It is the probability distribution over independent sets $I$ of a graph weighted proportionally to $\lambda^{|I|}$ with fugacity parameter $\lambda$. We prove that at the uniqueness threshold of the hardcore model on the $d$-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree $d… 195 Citations Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets • Mathematics 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science • 2011 This work refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion for establishing strong spatial mixing (and hence uniqueness) for the hard-core model, and provides a new lower bound for the uniqueness threshold. Improved mixing condition on the grid for counting and sampling independent sets • Mathematics • 2013 The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication Uniqueness of Gibbs measures for continuous hardcore models • Mathematics, Computer Science The Annals of Probability • 2019 We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter$\lambda > 0$. In this Decay of Correlations for the Hardcore Model on the$d$-regular Random Graph • Mathematics, Computer Science • 2014 The local weak limit of the hardcore model on random regular graphs is determined asymptotically until just below its condensation threshold, showing that it converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs • Mathematics, Computer Science ArXiv • 2020 The limiting distribution of the weights of the ordered and disordered phases at criticality is determined and exponential decay of correlations away fromcriticality is proved. Spatial mixing and the connective constant: optimal bounds • Mathematics, Computer Science SODA 2015 • 2015 The best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant are proved. Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model • Mathematics 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) • 2016 The hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter λ > 0 is studied, and it is proved that there exists a constant Δ<sub>0</sub> such that for all graphs with maximum degree Δ, the mixing time of the Glauber dynamics is O(nlog n) when λ <; λ<sub*c</sub>(Δ). Algorithmic Pirogov–Sinai theory • Mathematics Probability Theory and Related Fields • 2019 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 Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2 • Mathematics APPROX-RANDOM • 2013 An event is constructed that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours, and vastly improved bounds on the number of contours are obtained by relating them to a new class of self-avoiding walks on an oriented version of Z^2. Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region • Mathematics, Computer Science SODA • 2022 The 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. ## References SHOWING 1-10 OF 53 REFERENCES On the hardness of sampling independent sets beyond the tree threshold • Mathematics, Computer Science • 2007 It is conjecture that λc is in fact the exact threshold for this computational problem, i.e., that for λ >λc it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph. 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A new rapidly mixing Markov chain for independent sets is defined and a polynomial upper bound for the mixing time of the new chain is obtained for a certain range of values of the parameter ?, which is wider than the range for which the mixingTime of the Luby?Vigoda chain is known to be polynomially bounded.
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We consider Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that
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Proceedings 2001 IEEE International Conference on Cluster Computing
• 2001
If the relaxation time of the dynamics on trees and on certain hyperbolic graphs satisfies /spl tau//sub 2/=O(n), then the correlation coefficient, and the mutual information, between any local function, decays exponentially in the distance between the window and the boundary.
Algorithmic Barriers from Phase Transitions
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2008 49th Annual IEEE Symposium on Foundations of Computer Science
• 2008
It is proved that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi, and that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring, which means that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail.
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Proceedings of the National Academy of Sciences
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Empirical evidence suggests that local Monte Carlo Markov chain strategies are effective up to the clustering phase transition and belief propagation up toThe condensation point and refined message passing techniques (such as survey propagation) may also beat this threshold.