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5
votes
3answers
13k views

Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...
1
vote
0answers
45 views

Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$: $f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $ where $P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq ...
25
votes
1answer
2k views

When should we expect Tracy-Widom?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
0
votes
0answers
60 views

Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let \begin{equation} I = \int_D g(\textbf{x})d\textbf{x}, \end{equation} where $D \subset ...
15
votes
6answers
2k views

Can I derive the Boltzmann distribution by an invariance argument?

In statistical mechanics, the Boltzmann distribution gives the probability of a system being in state $i$ as $$\displaystyle \frac{e^{- \beta E_i}}{\sum_i e^{-\beta E_i}}$$ where $E_i$ is the energy ...
15
votes
2answers
1k 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. ...
16
votes
4answers
949 views

Pennies on a carpet problem

I recently read the following "open problem" titled "Pennies on a carpet" in "An Introduction To Probability and Random Processes" by Baclawski and Rota (page viii of book, page 10 of following ...
2
votes
0answers
37 views

TAP expression for entropy [closed]

This paper by Barton and Cocco: http://www.phys.ens.fr/~cocco/Art/articlejstat.pdf claims on page 17 (Formula (30)) an expression for the "high-temperature" entropy of an Ising model, given its ...
4
votes
1answer
209 views

What is known about first return times to Markov partitions for Anosov diffeomorphisms?

Consider an Anosov diffeomorphism $T: M \rightarrow M$ and a corresponding Markov partition $\mathcal{R}$ of $M$. For $x \in M$, let $\mathcal{R}(x)$ denote the element of $\mathcal{R}$ containing $x$ ...
3
votes
0answers
120 views

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 ...
5
votes
3answers
341 views

Ergodic theory: from Dynamics to Gibbs measure

I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure. I have a compact space $X$, a probability ...
5
votes
1answer
236 views

Importance of Ornstein's isomorphism theorem

"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows ...
25
votes
1answer
868 views

Bouncing a ball down the stairs

In a nutshell, the question is whether it can be faster to bounce a ball down an infinite flight of stairs than to bounce it down a ramp with the same slope. To be more specific: this is a $2$ ...
1
vote
0answers
42 views

Can the cavity method be used to analyze graphs with loops which are short?

In statistical physics, the cavity method can be regarded as a generalization of the Bethe-Peierls iterative method in tree-like graphs to the case of graphs with loops that are not too short. I would ...
5
votes
0answers
49 views

Full distribution of FPTs in random walks on graphs

There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ...
7
votes
1answer
250 views

Existence of Limiting Distribution for Moving Regions in Stat. Phys. Models

As the title (hopefully) suggests, I've been trying to prove (or disprove) the existence of a limiting distribution for a certain projection in a statistical physics model. I'll give the details of ...
1
vote
0answers
28 views

PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities: $$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$ $$ H = \sum_{k=0}^{M} 2 k \left[ ...
0
votes
0answers
94 views

Classic question on integer partitions (with distinct summands)

I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem: Denote $R(N,L)$ ...
1
vote
2answers
218 views

Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...
5
votes
1answer
186 views

Beraha numbers and zeros of the chromatic polynomial of planar graphs

Question: What is the largest Beraha number known to be an accumulation point of real zeros of the chromatic polynomial of planar graphs? Background: The Beraha numbers $B_n=2+2cos(2\pi/n), ...
1
vote
0answers
91 views

Stochastic Resonance in Infinite Dimensions

I'll ask this from the point of view of physics more than of theoretical mathematics. I'm searching for a mathematical discussion of stochastic resonance interpreted in a PDE sense. This is a good ...
3
votes
1answer
134 views

Average entropy of quantum system in bipartite pure state for finite temperature

[I got halfway through writing this when I found the paper that answers the question in (essentially) the affirmative. I'll post it anyways in case anyone is interested.] Background: If a random ...
5
votes
1answer
463 views

Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here. Basically, it is a ...
3
votes
1answer
131 views

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 ...
6
votes
2answers
485 views

Does the random Lorenz gas have a non-trivial diffusion coefficient?

For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If ...
7
votes
2answers
438 views

Does there exist a potential which realizes this strange quantum mechanical system?

I have done some courses on quantum mechanics and statistical mechanics in the past. Since I also do math, I wonder about converge issues which are usually not such a problem in physics. One of those ...
26
votes
2answers
902 views

Is this graph polynomial known? Can it be efficiently computed?

I am a physicist, so apologies in advance for any confusing notation or terminology; I'll happily clarify. To provide a minimal amount of context, the following graph polynomial came up in my research ...
2
votes
0answers
182 views

Random matrices whose limit gives exact Wigner surmise

Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write ...
6
votes
2answers
570 views

References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
7
votes
2answers
151 views

Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$. If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's ...
-2
votes
1answer
102 views

What are the formula of representation of quasicrystals and the law or mechanism of the formation [closed]

I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics ...
18
votes
2answers
1k views

An Entropy Inequality (generalized)

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability ...
8
votes
1answer
147 views

Exact enumerations from two-dimensional stat mech models

Exact enumerations corresponding to the dimer model on a hexagonal grid, the dimer model on a square grid, and the four-vertex (aka square ice) model on a square grid are known, namely: lozenge ...
1
vote
2answers
386 views

$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$

Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type. What are the Lipschitz functions with zero integral with respect to the measure $\mu?$ Clearly any ...
5
votes
0answers
232 views

Kasteleyn, Gessel-Viennot and eigenvalues

The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...
1
vote
0answers
532 views

What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
5
votes
0answers
290 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 ...
3
votes
3answers
2k views

The relations between the Perelman's entropy functional and notions of entropy from statistical mechanics

I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, and notions of ...
4
votes
0answers
210 views

Generalized Markov Processes on CW complexes of dimension > 1

Markov processes have a large variety of applications to physics and chemistry (as well as many other fields). Such processes are formulated on graphs, i.e., CW complexes of dimension one. It is ...
5
votes
0answers
178 views

Proving conformal invariance of a field theory by property of its stress energy tensor.

I have a question about proving conformal invariance of a field theory by property of its stress energy tensor. In physics there is argument that when the stress-energy tensor is traceless, ...
3
votes
0answers
603 views

On Perelman's paper

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written: Fix a closed manifold $M$ with a probability measure $m$, and suppose that our ...
7
votes
2answers
306 views

Examples of Slowly Mixing Chains in Statistics

This should probably be community wiki, but I don't know how to set that myself. I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
0
votes
0answers
226 views

softmax activation function with infinite support ?

Hi, How do we calculate the terms of a softmax activation function with an infinite support ? That is, finding the $\{p_i\}_i$ with $p_i = {{e^{q_i}} \over {\sum_{j=1}^\infty e^{q_j} }}$ (how to ...
1
vote
1answer
146 views

First order approximation of the current in ASEP

I am searching for an elementary proof of the first order approximation of the current of particles in ASEP (asymmetric simple exclusion process) and TASEP (totally asymetric). To avoid technical ...
12
votes
0answers
201 views

Ising model - phase transition vs rapid mixing

Consider a graph $G=(V,E)$ and Ising model on that graph, i.e. configuration space is $\Omega=${$-1,+1$}$^V$ and energy of a configuration $s \in \Omega$ is given by: $H(s) = -\beta \sum_{u \sim ...
3
votes
1answer
183 views

Ising entropy of a finite L_1 x L_2 lattice

We know the entropy per site of the 2-d Ising model from Onsager's solution. Has anybody also calculated the entropy for a finite rectangle of size L_1 x L_2 with periodic boundary conditions (i.e. on ...
2
votes
0answers
115 views

Does a certain Theorem on Boltzmann Distributions exist?

Suppose $X_n(z)$ is a sequence of random variables with a boltzmann distribution on $\{1,2,\dots n\}.$ That is $$P(X_n(z)=j)=\frac{c_{j,n} z^j}{F_n(z)}$$ where $F_n(z)=\sum_{j=1}^n c_{j,n}z^j$ is a ...
1
vote
0answers
123 views

Generating Conditional Random Graphs

Let $G(n,p)$ be the usual random graph on $n$ vertices with each edge existing independently with probability $p$ (no self loops , or double edges not are allowed). I would like to simulate the ...
8
votes
1answer
799 views

Correlation-Function for Random Graph Ising Model

For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes ...
4
votes
3answers
2k views

Statistical physics of string theory

Is there any connection between statistical physics and string theory, or a statistical interpretation of string theory, perhaps? I mean, the way electromagnetic forces and thermodynamic laws are ...