# Questions tagged [ising-model]

The Ising Model, introduced by the physicist Wilhelm Lenz (1920), is one of the most well-known models of Statistical Mechanics, used to explain the behavior of ferromagnets, but later found to have connections with many other models. Example of topics in the area include existence of phase transitions, asymptotic behavior of correlation functions, critical exponents, graphical representations, and properties of the pressure/free-energy function.

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### Roots of a family of polynomials forming shapes

Let $f$ be a smooth and strictly concave function on $[0,1]$, where $f(0)=f(1)=0$. Let $F_n(x)=\underset{k=0}{\overset{n} \sum } \exp(nf(\frac kn))x^k$. The roots of $F_n$ seems to form "shapes&...
85 views

### Bounds on the entropy of the 2D Ising model

I am interested in good estimators of (or analytical bounds on) the entropy $\mathsf{H}_\beta:=-\sum_{\mathbf{x}} P(\mathbf{x})\log_2(P(\mathbf{x}))$ of the two-dimensional Ising model (with no ...
73 views

### Generalized Ising Model

I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...
57 views

### Approximating a Distribution with an Ising Model/pairwise MRF

I want to know if there are any results on approximating a distribution with an Ising model/pairwise Markov Random Fields (MRFs). Formally, let $\mathcal{I}$ be the set of all Ising models/pairwise ...
120 views

### A combinatorial identity on even spanning subgraphs in the Erdös-Renyi random graph with relations to the Ising model

Let $x \in \lbrack 0,1 \rbrack$. Then for any finite graph $G$ consider the Erdös-Renyi random graph where we independently keep each of the edges with probability $x$. Denote the corresponding ...
134 views

### An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
154 views

### Ising model with zero external field - marginalization

The pmf of Ising model is considered as $p(\boldsymbol{x})=\frac{1}{Z(\theta)} exp\left\{ \underset{\left(s,t\right)\in E}{\sum\theta_{st}}x_{s}x_{t}\right\},\quad \boldsymbol{x}\in \{-1,1\}^n$, where ...
128 views

### A proof for this equivalent version of the Infrared Bound/Gaussian Domination

I have recently asked this question in Physics Stackexchange, but as there was no success there, a friend pointed out that I might have a better shot here. Consider the Ising Model in the $d$-...
317 views

### Ising model, phase transition

What is the temperature for the phase transition in the triangular-lattice Ising model? and in the hexagonal-lattice Ising model?
136 views

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### Ising model: probability of a long path of minus under plus boundary conditions

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions. Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...
3k views

### 2d Ising model in conformal fields theory and statistical mechanics

I am not completely sure that this question is appropriate for this mathematical site. But since in the past I did get on MO couple of times nice answers to rather physical questions, I will try. ...
514 views

### Elementary proof of lack of phase transition in Ising models with external fields

I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the ...
652 views

### Uniqueness of Gibbs Measure on Ising model

If I understood this correctly, the Gibbs Specification for the Ising model on $ℤ^d$ dos not have a unique Gibbs Measure for β above the critical level. But what about the Ising model on a finite ...
Hello, The Hubbard-Stratonovich transformation $\exp(x^2) = \frac{1}{\sqrt{4 \pi}} \int_{-\infty}^{+\infty} du \exp(-\frac{u^2}{4} - xu)$ allows one to wirte the exponential of a the square of a ...