Questions tagged [gibbs-measure]

Questions on the Gibbs measure in any measurable space $(\mathbb{E}^\mathbb{T},\mathcal{F})$ defined by a family of potentials $\Phi=\{\Phi_t\}_{t\in\mathbb{T}}$ in a net $\mathbb{T}$. Here $\mathbb{E}$ is a topological space or a measurable space.

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Ising Model using morphological filters

How can i prove that a simple MRF of some interest in image processing and analysis is the Ising model, whose energy function is: $U(X)=a|X|+b_{1}|(X \ominus B_{1})|+b_{2}|(X \ominus B_{2})|$ when we ...
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Negative Log-Likelihood Loss with Gibbs distribution for beta approaching infinity

TL;DR: What happens with Gibb's distribution when $\beta \to \infty $ and why? $$ \lim_{\beta \to \infty} \frac{\exp(-\beta E(W, Y^i, X^i))}{\int_y \exp(-\beta E(W, y, X^i)) } \ = \ ? $$ Full ...
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13 views

Integral with function in the exponent / Gibbs Distribution via MaxEnt

In the context of a maximum entropy procedure, I am trying to evaluate the following definite integrals—but they have an arbitrary function $g(x)$ in the exponent : $\int\limits_0^\infty e^{-\lambda ...
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57 views

Define the Gibbs-Boltzmann distribution over vertices

Maybe my question is wrong or not clear so I would be grateful for any modification. I am discovering the Gibbs-Boltzmann distribution but it seems strange for me and really hard to understand! ...
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36 views

Examples of functions summables in $\mathbb{Z}^d$.

I know that $f_0(x)={ 1 \over \vert \vert x\vert \vert^\alpha}$, $f_0(0)=1$ , $\alpha>d$, $d\in \mathbb{N}$ is summable in $\mathbb{Z}^d$, i.e. $$ \sum_{x \in \mathbb{Z}^d} f(x)<\infty. $$ I ...
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1answer
482 views

Examples of graphs that are amenable and non-amenable

The amenable graph $G=(V, E)$ is a graph that satisfies the following $$ \inf\limits_{K \subset V,\, |K|< \infty} \frac{\partial K}{|K|}=0$$ I know for example that $\mathbb{Z}^2$ is amenable ...
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143 views

Getting a bound for Gibbs distribution mean

Suppose $F$ is a strictly convex and increasing function, $U$ a random variable with support $[0,1]$ and density $$ f_U(u)= \frac{e^{-\frac{1}{T}F(u)}}{\int_{0}^{1} e^{-\frac{1}{T} F(x)} dx}.$$ Do we ...
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1answer
109 views

Ergodicity of $\mu^0_\beta$ on a particular $\sigma$-algebra (Ising Model)

Consider the Ising Model on $\mathbb{Z}^d$ with nearest neighbors interaction, free boundary condition,$h=0$,and $\beta>0$. I would like to prove that for all local functions $f$ and $g$ such that ...
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157 views

A correlation inequality of the Ising Model

In the Ising Model with $+$ boundary condition in dimension $2$, but possibly one could ask about dimension $d$. Set $\Lambda_N:=[-N,N]^2$, let $\beta >0$ be the inverse of the temperatura and $h&...
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80 views

Compute transition probabilities for reversible dynamics wrt Gibbs measure

I'm looking at this paper about Gibbs measure on random graph. I don't understand how to compute transition probabilities for the first example (known as Glauber or Metropolis dynamics, pag 4-5) for ...
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189 views

Uniqueness of Gibbs measure for rotator model in one dimension

I am trying to solve a problem in a course of Y. Velenik (models with continuous symmetry, exercice 8.18: http://www.unige.ch/math/folks/velenik/smbook/index.html): Show that in dimension $d=1$ ...
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83 views

Ising Model on 2k-regular graphs

Is Ising model on any infinite $2k$-regular graph (where the vertex degree is exactly $2k$) equal to Ising model on $\mathbb{Z}^k$ ($\mathbb{Z}^k$ lattice) ( where the vertex degree is $2k$ as well ...
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722 views

How to Derive Gibbs Sampling Update Formula for Hidden Markov Model?

I want to understand how to derive the update formula for Gibbs sampling for Hidden Markov Model, for example, in here: $$p(z_t | \mathbf{x}, \mathbf{z}_{\setminus t}, \boldsymbol{\alpha}, > \...
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114 views

Limit of correlation function using transfer-matrix method

This question is about a stochastic process theory. I really very bad in this topic. That's why I have to ask for help. I may mistranslate some terms but I'll do my best to give you right information. ...
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77 views

Conditions for the ground state of Gibbs ensemble not to be “degenerate”

I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions. I am looking for books or other resources ...