Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions

  title={Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions},
  author={Jingcheng Liu and Alistair Sinclair and Piyush Srivastava},
  journal={SIAM Journal on Computing},
We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $\Delta$-regular tree) implies strong spatial mixing on all graphs of maximum degree $\Delta$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region… 

Absence of zeros implies strong spatial mixing

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Polynomial-Time Approximation of Zero-Free Partition Functions

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Towards derandomising Markov chain Monte Carlo

A new framework to derandomise certain Markov chain Monte Carlo algorithms, namely efficient deterministic approximate counting algorithms for hypergraph independent sets and hypergraph colourings, under local lemma type conditions matching, up to lower order factors, their state-of-the-art randomised counterparts.

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In this paper we prove that for any integer q ≥ 5 , the anti-ferromagnetic q -state Potts model on the infinite ∆ -regular tree has a unique Gibbs measure for all edge interaction parameters w ∈ [1 −



Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

The results of this paper indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems on graphs of maximum degree $$d$$d for parameters outside the uniqueness region.

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This work introduces the contraction property as a unified sufficient condition to devise FPTAS via either Weitz's algorithm or Barvinok's algorithm and shows the existence of correlation decay in these regions based on the zero-freeness of the partition function.

Location of zeros for the partition function of the Ising model on bounded degree graphs

The location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$ is studied, both in the ferromagnetic and the anti-ferromagnetic case, exactly as a function of the inverse temperature and the degree.

Approximate counting via correlation decay in spin systems

A potential method is used to analyze the amortized behavior of this correlation decay and establish a correlation decay that guarantees an inverse-polynomial precision by polynomial-time local computation, which gives an FPTAS for spin systems on arbitrary graphs.

Correlation Decay up to Uniqueness in Spin Systems

It is shown that a two-state anti-ferromagnetic spin system exhibits strong spatial mixing on all graphs of maximum degree at most Δ if and only if the system has a unique Gibbs measure on infinite regular trees of degree up to Δ, where Δ can be either bounded or unbounded.

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the

Strong spatial mixing for lattice graphs with fewer colours

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Improved FPTAS for Multi-spin Systems

The deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function for a class of multi-spin systems is designed and an FPTAS for the Potts models with inverse temperature β up to a critical threshold is given, confirming a conjecture in [10].

Improved bounds for sampling colorings

  • Eric Vigoda
  • Mathematics
    40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
  • 1999
The main result is the design Markov chain that converges in O(nk log n) time to the desired distribution when k>11/6 /spl Delta/.

Combinatorics and Complexity of Partition Functions