# 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} }

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

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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.

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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…

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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…

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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.

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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.

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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.

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- Mathematics2016 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).

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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.

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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.

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