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.