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Approximate Counting via Correlation Decay in Spin SystemsPowerPoint Presentation

Approximate Counting via Correlation Decay in Spin Systems

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### 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.

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