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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Can nonnegative functions $f(x,y,z)$ be written as a product of pairwise functions $u(x,y) v(y,z) w(x, z)$?

In my course on probabilistic graphical models, my professor made a claim which I find a little sus. In discussing the equivalence between Markov Random Fields and Factor Graphs, the following example ...
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Training an energy-based model (EBM) using MCMC

I'm reading this paper about training energy-based models (EBMs) and don't understand the parameters that we are training for? The part that is relevant to the question is in pages 1-4. Here is the ...
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What are the coefficients of this partition function in the following Ising model?

Investigating further questions around this question: Example of sequence of graphs which satisfy the Riemann hypothesis? leads to the partition function $Z$ of the Ising model of the graph defined ...
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Example of sequence of graphs which satisfy the Riemann hypothesis or the prime number theorem?

Let us look at the sequence of bipartite graphs $G_n = (V_n, E_n)$ where $V_n = A_n \cup B_n$ defined in this quesiton: Why is this bipartite graph a partial cube, if it is? . The shortest path ...
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Interesting and surprising applications of the Ising Model

One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in ...
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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&...
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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 ...
110 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 ...
1 vote
70 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 ...
1 vote
147 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 ...
1 vote
139 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 (...
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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 ...
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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 $d$-...
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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?
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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 ...
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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. ...
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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 ...
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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 ...
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Hubbard-Stratonovich Transformation

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 ...
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