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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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 ...
mathoverflowUser's user avatar
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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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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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The uniform odd and even subgraph of $\mathbb{Z}^2$

Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
Frederik Ravn Klausen's user avatar
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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 ...
James's user avatar
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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 ...
Arvind Rameshwar's user avatar
3 votes
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Hamiltonian on the torus

In discrete models like Ising we have Hamiltonians of the form $$H(\sigma)=\frac{1}{N}\sum_{i=1}^{N}J_{ij}\sigma_{i}\sigma_{j},$$ where $\sigma_{i}=\pm 1$ , $J_{ij}$ interaction coefficients and N ...
user133100's user avatar
2 votes
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The fluctuations of a random path

Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
Frederik Ravn Klausen's user avatar
2 votes
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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 ...
Garfield's user avatar
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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 $d$-...
Kernel's user avatar
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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 ...
Uthsav Chitra's user avatar
1 vote
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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 (...
Min Wu's user avatar
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Random Cluster Model only for bond percolation?

Can someone please tell me which of the following statements I make are true of the current state of the art: The Random Cluster Model is a generalization of bond percolation (with possibly different ...
Jeff's user avatar
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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 ...
Giovanni Febbraro's user avatar