Correlation decay and the absence of zeros property of partition functions

Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok \cite{barvinok2017combinatorics} for designing deterministic approximation algorithms for various graph counting and computing partition functions problems. Earlier methods for solving the same problem include the one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply sometimes coincide or nearly coincide. In this paper we show that this is more than just a coincidence. We establish that if the interpolation method is valid for a family of graphs satisfying the self-reducibility property, then this family exhibits a form of correlation decay property which is asymptotic Strong Spatial Mixing (SSM) at distances $\omega(\log^{3}n)$, where $n$ is the number of nodes of the graph. This applies in particular to amenable graphs such as graphs which are finite subsets of lattices. Our proof is based on a certain graph polynomial representation of the associated partition function. This representation is at the heart of the designing the polynomial time algorithms underlying the interpolation method itself. We conjecture that our result holds for all, and not just amenable graphs.

[1]  J. Jonasson.  Uniqueness of uniform random colorings of regular trees , 2002 .

[2]  M. Jerrum.  Counting, Sampling and Integrating: Algorithms and Complexity , 2003 .

[3]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[4]  Péter Csikvári,et al.  Benjamini-Schramm continuity of root moments of graph polynomials , 2012, Eur. J. Comb..

[5]  Dana Randall,et al.  Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2 , 2012, APPROX-RANDOM.

[6]  David Gamarnik,et al.  Counting without sampling: new algorithms for enumeration problems using statistical physics , 2006, SODA '06.

[7]  Alexander I. Barvinok,et al.  Computing the Partition Function for Cliques in a Graph , 2014, Theory Comput..

[8]  Dmitriy Katz,et al.  Strong spatial mixing of list coloring of graphs , 2012, Random Struct. Algorithms.

[9]  Piyush Srivastava,et al.  A Deterministic Algorithm for Counting Colorings with 2-Delta Colors , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[10]  Will Perkins,et al.  Algorithmic Pirogov–Sinai theory , 2018, Probability Theory and Related Fields.

[11]  Han Peters,et al.  On a conjecture of Sokal concerning roots of the independence polynomial , 2017, Michigan Mathematical Journal.

[12]  Alexander I. Barvinok.  Computing the partition function of a polynomial on the Boolean cube , 2015, ArXiv.

[13]  Allan Sly,et al.  Computational Transition at the Uniqueness Threshold , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[14]  Stan Zachary,et al.  Markov random fields and Markov chains on trees , 1983 .

[15]  Thomas P. Hayes,et al.  Improved Strong Spatial Mixing for Colorings on Trees , 2019, APPROX-RANDOM.

[16]  Liang Li,et al.  Correlation Decay up to Uniqueness in Spin Systems , 2013, SODA.

[17]  Pinyan Lu,et al.  Improved FPTAS for Multi-spin Systems , 2013, APPROX-RANDOM.

[18]  Dror Weitz,et al.  Counting independent sets up to the tree threshold , 2006, STOC '06.

[19]  Alexander I. Barvinok,et al.  Combinatorics and Complexity of Partition Functions , 2017, Algorithms and combinatorics.

[20]  Viresh Patel,et al.  Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials , 2016, Electron. Notes Discret. Math..

[21]  Will Perkins,et al.  Algorithms for #BIS-hard problems on expander graphs , 2018, SODA.

[22]  Jan Vondrák,et al.  Computing the Independence Polynomial: from the Tree Threshold down to the Roots , 2016, SODA.

[23]  Frank Kelly,et al.  Stochastic Models of Computer Communication Systems , 1985 .

[24]  Dmitriy Katz,et al.  Correlation decay and deterministic FPTAS for counting list-colorings of a graph , 2007, SODA '07.

[25]  F. Papangelou.  GIBBS MEASURES AND PHASE TRANSITIONS (de Gruyter Studies in Mathematics 9) , 1990 .

[26]  David Gamarnik,et al.  Simple deterministic approximation algorithms for counting matchings , 2007, STOC '07.

[27]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[28]  F. Spitzer.  Markov Random Fields on an Infinite Tree , 1975 .

[29]  Will Perkins,et al.  Algorithms for #BIS-Hard Problems on Expander Graphs , 2020, SIAM J. Comput..

[30]  Jean-Pierre Tignol,et al.  Galois' theory of algebraic equations , 1988 .

[31]  D. Gamarnik,et al.  Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models , 2008 .

[32]  R. Dobrushin,et al.  Completely analytical interactions: Constructive description , 1987 .

[33]  Alexander I. Barvinok,et al.  Approximating real-rooted and stable polynomials, with combinatorial applications , 2018, ArXiv.

[34]  Dmitriy Katz,et al.  Correlation decay and deterministic FPTAS for counting colorings of a graph , 2012, J. Discrete Algorithms.

[35]  Alexander I. Barvinok,et al.  Computing the Permanent of (Some) Complex Matrices , 2014, Foundations of Computational Mathematics.