The paper studies the location of the zeros of the Ising model partition function ZIsing(G; b), which are indicative of possible phase transitions in the model. The authors establish a new, near-optimal zero-free disk for ZIsing(G; b) that is larger than previously known regions.
The key results are:
The authors show that for any positive integer Δ and any graph G of maximum degree at most Δ, ZIsing(G; b) ≠ 0 for all b ∈ C satisfying |b-1/b+1| ≤ 1-o(1)/(Δ-1), where o(1) → 0 as Δ → ∞. This is optimal in the sense that 1-o(1)/(Δ-1) cannot be replaced by c/(Δ-1) for any constant c > 1, subject to a complexity-theoretic assumption.
The authors also show that if G additionally has girth at least g, then ZIsing(G; b) ≠ 0 for any b in a larger disk with radius 1-ε/(Δ-1), for any ε ∈ (0, 1).
The proof uses a standard reformulation of the Ising partition function as the generating function of even sets, and establishes a zero-free disk for this generating function inspired by techniques from statistical physics on partition functions of polymer models. The approach is quite general and the authors discuss extensions to certain types of polymer models.
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by Viresh Patel... at arxiv.org 04-24-2024
https://arxiv.org/pdf/2311.05574.pdfDeeper Inquiries