A near-optimal zero-free disk for the Ising model
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917647828910080 |
|---|---|
| author | Patel, Viresh Regts, Guus Stam, Ayla |
| author_facet | Patel, Viresh Regts, Guus Stam, Ayla |
| contents | The partition function of the Ising model of a graph $G=(V,E)$ is defined as $Z_{\text{Ising}}(G;b)=\sum_{σ:V\to \{0,1\}} b^{m(σ)}$, where $m(σ)$ denotes the number of edges $e=\{u,v\}$ such that $σ(u)=σ(v)$. We show that for any positive integer $Δ$ and any graph $G$ of maximum degree at most $Δ$, $Z_{\text{Ising}}(G;b)\neq 0$ for all $b\in \mathbb{C}$ satisfying $|\frac{b-1}{b+1}| \leq \frac{1-o_Δ(1)}{Δ-1}$ (where $o_Δ(1) \to 0$ as $Δ\to \infty$). This is optimal in the sense that $\tfrac{1-o_Δ(1)}{Δ-1}$ cannot be replaced by $\tfrac{c}{Δ-1}$ for any constant $c > 1$ subject to a complexity theoretic assumption.
To prove our result we use a standard reformulation of the partition function of the Ising model as the generating function of even sets. We establish a zero-free disk for this generating function inspired by techniques from statistical physics on partition functions of a polymer models. Our approach is quite general and we discuss extensions of it to a certain types of polymer models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05574 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A near-optimal zero-free disk for the Ising model Patel, Viresh Regts, Guus Stam, Ayla Combinatorics Discrete Mathematics Data Structures and Algorithms Mathematical Physics The partition function of the Ising model of a graph $G=(V,E)$ is defined as $Z_{\text{Ising}}(G;b)=\sum_{σ:V\to \{0,1\}} b^{m(σ)}$, where $m(σ)$ denotes the number of edges $e=\{u,v\}$ such that $σ(u)=σ(v)$. We show that for any positive integer $Δ$ and any graph $G$ of maximum degree at most $Δ$, $Z_{\text{Ising}}(G;b)\neq 0$ for all $b\in \mathbb{C}$ satisfying $|\frac{b-1}{b+1}| \leq \frac{1-o_Δ(1)}{Δ-1}$ (where $o_Δ(1) \to 0$ as $Δ\to \infty$). This is optimal in the sense that $\tfrac{1-o_Δ(1)}{Δ-1}$ cannot be replaced by $\tfrac{c}{Δ-1}$ for any constant $c > 1$ subject to a complexity theoretic assumption. To prove our result we use a standard reformulation of the partition function of the Ising model as the generating function of even sets. We establish a zero-free disk for this generating function inspired by techniques from statistical physics on partition functions of a polymer models. Our approach is quite general and we discuss extensions of it to a certain types of polymer models. |
| title | A near-optimal zero-free disk for the Ising model |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms Mathematical Physics |
| url | https://arxiv.org/abs/2311.05574 |