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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.

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.

23 questions
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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 have a known bound for $\Pr\{U>y\}$ for any $y>0$…
rookie
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5
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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 but in special way: only vertices with distance one…
Toni
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Proving an inequality associated with Gibbs measure

In the statistical physics research, I encountered the following inequality that I need to prove: \begin{aligned} & \int_{[0,1]^n} \frac{\partial f(a; x_1, \cdots, x_n)} {\partial a}\mathrm{d}x_1\cdots \mathrm{d}x_n \int_{[0,1]^n} f^2(a; x_1,…
wangfulyh
  • 31
3
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1 answer

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 and the Bethe lattice (Cayley tree) is…
Jir
  • 59
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1 answer

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 $d=1$ there is a unique gibbs measure for the $O(N)$ model…
Nathan Noiry
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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}, > \boldsymbol\beta) \propto \dfrac{C_{x_t, z_t}^{-t} +…
ihoho
  • 31
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1 answer

Characterizing Gibbs measures

I am self-studying spin glasses and the first exercise I want to do is the following: Characterize probability measures on $\{-1, +1\}^N$ that arise as Gibbs measures for a Hamiltonian. I am not exactly sure what to do here. Since it’s the first…
MathIsLife12
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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 question I asked this on stats.stackexchange before but…
rubencart
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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 $f(-\sigma)=f(\sigma)$ and $g(-\sigma)=g(\sigma)$,…
Kernel
  • 1,951
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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 $h>0$ be the magnetic field. For $x,y \in \Lambda_N$, is…
Kernel
  • 1,951
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Relation between mean and $\beta$ parameter of discrete Boltzmann distribution with degeneracies

I am trying to compute the mean $\bar{E}$ of a Boltzmann distribution $$ p(\pmb{x}) = \frac{1}{Z} e^{-\beta E(\pmb{x})}, $$ knowing that there is a total of $N$ microstates $\pmb{x}$ and $[\alpha N]$ energy levels, and that each energy level has…
edfi
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Surface order large deviation in Ising ferromagnet

Background: A familiar behaviour of independent and identically distributed (i.i.d.) random variables $X_1, X_2,\ldots X_n$ is concentration: the probability that the sum $X_1+X_2+\ldots X_n$ exceeds $\mathbb{E}(X_1+X_2+\ldots X_n)$ by…
anurag anshu
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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 that cover the topic of this subsection in detail…
Szabolcs
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Why do we use 0.23 as the acceptance rate for the Metropolis-Hastings algorithm?

I understand that traditionally the value of 0.23 is suggested for the acceptance rate when monitoring a Metropolis-Hastings algorithm, why is this and what would the effect of having a higher acceptance rate be?
MathewG
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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! Generally, according to the EXPONENTIAL RANDOM GRAPHS,…
Noah16
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