Approximate Counting via Correlation Decay in Spin Systems

1. Approximate Counting via Correlation Decay in Spin Systems Pinyan Lu Microsoft Research Asia Joint with Liang Li (Peking University) Yitong Yin (Nanjing University)

2. Spin Systems • System and spin states • Configuration • Edge function • Vertex function b: [q]-> • Weight of a configuration • Partition function:

3. 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 .

4. 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 , .

5. 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.

6. 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

7. Two-State Spin System • After normalization: , • Anti-ferromagnetic system: • Hardcore model : • Ising model:

8. 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.

9. 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.

10. 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.

11. 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?

12. 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.

14. 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.

15. Open Questions • Hardness result when the uniqueness does not hold. • Multi-spin systems? • Application of the approach to other counting problems.