Core Concepts

The Ising model partition function ZIsing(G; b) is non-zero for all graphs G of maximum degree at most Δ and all complex values of the edge interaction parameter b that satisfy |b-1/b+1| ≤ 1-o(1)/(Δ-1), which is near-optimal.

Abstract

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

Deeper Inquiries

The near-optimal zero-free disk for the Ising partition function has significant implications for the computational tractability of estimating the partition function. The zero-free disk ensures that the partition function of the Ising model does not vanish within a certain region of the complex plane, allowing for efficient computation of the partition function within that region. This is crucial for computational algorithms that rely on the partition function, as it guarantees that the computation will not encounter zeros that could lead to inaccuracies or computational challenges. By establishing a region where the partition function is non-zero, the near-optimal zero-free disk enables the development of more efficient algorithms for estimating the Ising partition function on graphs of bounded maximum degree.

Improving the zero-free disk, especially in the imaginary direction, could have significant implications for quantum computing applications. In quantum mechanics and quantum computing, the Ising model plays a crucial role in understanding quantum phase transitions and quantum algorithms. By extending the zero-free region in the imaginary direction, it may be possible to enhance the computational tractability of approximating the partition function for non-real values of the parameter, which are relevant in quantum computing settings. This improvement could lead to the development of more efficient quantum algorithms for approximating the Ising partition function and studying quantum phase transitions.

The general approach of using block structure and the polymer method to establish zero-free regions for graph polynomials and partition functions can be applied to various other important graph invariants and combinatorial models. By considering the block structure of subgraphs and utilizing techniques from statistical physics, it is possible to analyze the behavior of partition functions and generating functions for a wide range of graph models. This approach can be extended to study the zero-free regions of other graph polynomials, such as the Tutte polynomial, chromatic polynomial, and homomorphism densities. By leveraging the block structure and polymer method, researchers can explore the computational tractability and phase transitions of various combinatorial models on graphs.

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