Approximate Counting via Correlation Decay in Spin Systems
Pinyan Lu Microsoft Research Asia
Joint with Liang Li (Peking University)
Yitong Yin (Nanjing University)
Spin Systems
• System and spin states • Configuration • Edge function • Vertex function b: [q]-> • Weight of a configuration
• Partition function:
Gibbs Measure
• is a distribution over all configurations. • We can define the marginal distribution of
spins on a vertex . • We can also fix the configuration of a subset of
the vertices as , and define the conditional distribution of other vertex as .
Weak Correlation Decay
A spin system on a family of graphs is said to have exponential correlation decay if for any graph G=(V,E) in the family, any and , .
Correlation Decay
A spin system on a family of graphs is said to have exponential correlation decay if for any graph G=(V,E) in the family, any and , , where is the subset on which and differ.
Tree Uniqueness Thresholds
• Spin system on infinite regular graph: Grid, infinite regular tree.
• Uniqueness of the Gibbs Measure
• Equivalent to the weaker correlation decay on that infinite regular graph
Two-State Spin System
• After normalization: , • Anti-ferromagnetic system:
• Hardcore model : • Ising model:
Hardcore Model [Weitz 2006]
• Strong correlation decay holds on all graphs with maximum degree at most iff the uniqueness condition holds on infinite -regular tree.
• Self Avoiding walk(SAW) tree: transform a general graph to a tree and the keep the marginal distribution for the root.
• Monotonicity: any tree with degree at most decays at least as fast as the complete -regular tree.
Our Results
• The system is of correlation decay on all the graphs with maximum degree iff the system exhibits uniqueness on all the infinite regular trees up to degree .
• In particular, if the system exhibits uniqueness on infinite regular trees of all degrees, then the system is of correlation decay on all graphs.
Our Results
• We obtain a FPTAS as long as the system satisfies the uniqueness condition.
• Almost all the previous algorithmic results for two-state anti-ferromagnetic spin systems can be viewed as corollaries of our result by restricting some of the parameters.
• Moreover, in most of the cases, even in such restricted setting, our results no only covers but also improves the previous best results.
From correlation decay to FPTAS
• Marginal distribution -> partition function
• Correlation decay-> estimate by a local neighborhood: depth of the SAW tree.
• How about unbounded degree?
Computational Efficient Correlation Decay
• M-based depth: – ;– , if is one of the children of .
• Exponential correlation decay with respect to M-based depth.
• Computational efficient correlation decay supports FPTAS for general graph.
Proof Sketch for Correlation Decay
• Self avoiding walk tree and recursion relation on tree:
• Estimate the error for one recursive step:
• Use a potential function to amortize it.
Open Questions
• Hardness result when the uniqueness does not hold.
• Multi-spin systems?
• Application of the approach to other counting problems.