# Fisher zeros and correlation decay in the Ising model

@inproceedings{Liu2018FisherZA, title={Fisher zeros and correlation decay in the Ising model}, author={Jingcheng Liu and Alistair Sinclair and Piyush Srivastava}, booktitle={Information Technology Convergence and Services}, year={2018} }

The Ising model originated in statistical physics as a means of studying phase transitions in magnets, and has been the object of intensive study for almost a century. Combinatorially, it can be viewed as a natural distribution over cuts in a graph, and it has also been widely studied in computer science, especially in the context of approximate counting and sampling. In this paper, we study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the…

## 28 Citations

### Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions

- MathematicsSIAM Journal on Computing
- 2022

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs…

### Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

- MathematicsForum of Mathematics, Sigma
- 2022

P-hardness for approximating the partition function on graphs of maximum degree $\Delta$ when $b$, the edge-interaction parameter, is in the interval $(0,\frac{\Delta-2}{\Delta}]$ and $\lambda$ is a non-real on the unit circle.

### Approximation algorithms for the random-field Ising model

- Mathematics, Computer ScienceArXiv
- 2021

This work establishes the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and ∆.

### UvA-DARE (Digital Academic Repository) Location of zeros for the partition function of the Ising model on bounded degree graphs

- Mathematics
- 2020

The seminal Lee–Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in C . In fact, the union of the zeros of all graphs…

### The complexity of approximating the complex-valued Potts model

- Mathematicscomputational complexity
- 2022

We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the…

### Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

- Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

This proof utilizes that the graph partition function is a divisor of the partition function for Weitz's self-avoiding walk tree and leads to new tools for the analysis of the influence of vertices, and may be of independent interest for the study of complex zeros.

### Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

- MathematicsJournal of Statistical Physics
- 2021

This work introduces the contraction property as a unified sufficient condition to devise FPTAS via either Weitz's algorithm or Barvinok's algorithm and shows the existence of correlation decay in these regions based on the zero-freeness of the partition function.

### Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

- Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

It is shown that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold, improving the quasi-polynomial running time of Weitz's deterministic correlation decay algorithm.

### More on zeros and approximation of the Ising partition function

- Mathematics, Computer ScienceForum of Mathematics, Sigma
- 2021

The method of polynomial interpolation is applied, for which it is proved that $\sum _x e^{\tilde f}(x)} \ne 0$ for complex-valued polynomials $\tilde {f}$ in a neighborhood of a real-valued f satisfying the above mentioned conditions.

### Efficient learning of ground&thermal states within phases of matter

- Mathematics
- 2023

It is shown it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable's locality -- an exponential improvement.

## References

SHOWING 1-10 OF 35 REFERENCES

### The Ising Partition Function: Zeros and Deterministic Approximation

- Mathematics2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

A tight version of the Lee-Yang theorem is established for the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms were known for a wide range of parameters.

### Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model

- Physics, Mathematics
- 1952

The problems of an Ising model in a magnetic field and a lattice gas are proved mathematically equivalent. From this equivalence an example of a two-dimensional lattice gas is given for which the…

### Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

- MathematicsSODA
- 2012

The results of this paper indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems on graphs of maximum degree $$d$$d for parameters outside the uniqueness region.

### Counting without sampling: Asymptotics of the log‐partition function for certain statistical physics models

- MathematicsRandom Struct. Algorithms
- 2008

It is proved that asymptotically the logarithm of the number of independent sets of any r‐regular graph with large girth when rescaled is approximately constant if r ≤ 5, as the size of the underlying graph goes to infinity.

### Density of the Fisher Zeroes for the Ising Model

- Physics
- 2001

The density of the Fisher zeroes, or zeroes of the partition function in the complex temperature plane, is determined for the Ising model in zero field as well as in a pure imaginary field iπ/2.…

### Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

- Computer Science, MathematicsQuantum
- 2019

A deterministic polynomial-time approximation scheme for the Ising model partition function when the interactions and external fields are absolutely bounded close to zero and it is proved that for this class of Ising models the partition function does not vanish.

### The Ising model and percolation on trees and tree-like graphs

- Physics
- 1989

We calculate the exact temperature of phase transition for the Ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field. In the same setting, we calculate…

### Approximating partition functions of the two-state spin system

- MathematicsInf. Process. Lett.
- 2011

### Correlation Decay up to Uniqueness in Spin Systems

- MathematicsSODA
- 2013

It is shown that a two-state anti-ferromagnetic spin system exhibits strong spatial mixing on all graphs of maximum degree at most Δ if and only if the system has a unique Gibbs measure on infinite regular trees of degree up to Δ, where Δ can be either bounded or unbounded.