Zeros of the Partition Function of the Ising Model on Bounded Degree Graphs
Talkat Deterministic Counting, Probability, and Zeros of Partition Functions (Simons Institute workshop, 2019) · 2019-03-18..03-22
summary
Proves explicit zero-free interval on the unit circle for the ferromagnetic Ising partition function on bounded-degree graphs by analyzing tree recursions via Möbius transformation and expansion on trees and tools from complex dynamics; extends the argument to multivariate interactions, gives a quantitative bound depending on degree and interaction parameter, and uses that absence of zeros to derive a deterministic FPTAS for counting in those parameter regions. Paper outlines open problems about the maximal zero-free domain.
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chapters (16)
- 0:00Introduction
- 1:00Partition function basics
- 3:30Zeros of partition function
- 6:02Known results overview
- 8:14Open problems
- 14:27Partition function applications
- 17:20Complex dynamics on trees
- 22:27Möbius transformations analysis
- 24:13Complex dynamics fundamentals
- 27:20Critical parameters and fixpoints
- 30:57Parameter tita and critical values
- 33:28Dynamical systems and Julia sets
- 36:17Multi-variable extensions
- 38:28Open problems and future
- 43:49Computational challenges
- 44:32Q&A