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21
votes
1answer
1k 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
16 views

Can Cavity method to analyze graph with loops that are short?

In statistical physics,Cavity method can be regarded as a generalization of the Bethe Peierls iterative method in tree-like graphs to the case of graph with loops that are not too short. And I want ...
5
votes
1answer
397 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. ...
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 ...
11
votes
0answers
372 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 ...
9
votes
2answers
263 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. ...
22
votes
1answer
443 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 $2$ ...
8
votes
1answer
105 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
364 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 ...
6
votes
2answers
400 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 ...
0
votes
0answers
76 views

Fitting distribution to spatial data

I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below. ...
3
votes
0answers
131 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
223 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 ...
4
votes
0answers
192 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
771 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 ...
0
votes
0answers
93 views

Master Equation to Fokker-Planck for a Jump-Diffusion

Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
3
votes
0answers
179 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
97 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
522 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 ...
6
votes
2answers
286 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
131 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
130 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 ...
9
votes
0answers
154 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
164 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
2answers
5k 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 ...
2
votes
0answers
108 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 ...
0
votes
0answers
108 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
594 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
1k 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 ...
23
votes
0answers
1k views

Are there lots of integer homology three-spheres?

The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and ...
13
votes
1answer
414 views

Arctic regions in higher dimensional zonotopes

Same way as the two dimensional tilings by rhombi come from minimal surfaces in a $D$ dimensional cubical lattice as mentioned in this answer, one can consider higher dimensional zonotopes tiled by ...
8
votes
1answer
262 views

computing average height-functions for lozenge tilings

Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? I prefer to ...
29
votes
10answers
3k views

Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...
16
votes
1answer
377 views

How much universality is there for contact processes?

A couple of weeks ago I had to pick my daughter up from her nursery because of suspected chicken pox. It turned out to be a false alarm, but while I was waiting at the doctor's surgery to establish ...
9
votes
1answer
1k views

Wick rotation and the Riemann zeta function

The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations. Background I have by now ...
10
votes
7answers
978 views

Probabilistic (and other mathematical) methods of physics without the physics?

Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually ...
2
votes
1answer
315 views

Are there any physical phenomena of the heat transfer critically depending on diffusion coefficient?

Hello, I am considering the following non-linear heat equation $$ \left(\frac{\partial}{\partial t}-\nu\: \Delta \right) u(t,x) = F(t,x) \sigma(u(t,x)),\qquad (t,x)\in R_+\times R^d $$ where ...
8
votes
1answer
1k views

Entropy of the Ising model

Consider the standard Ising model on $[0,N]^2$ for $N$ large. By that I mean the square-lattice Ising model without external field, inside an $N$-by-$N$ square. What is its entropy for $N$ large? It ...
1
vote
1answer
196 views

Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance

I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations ...
16
votes
3answers
746 views

Ising model on groups

Can anything interesting be deduced about the properties of a group from the behavior of the Ising model on its Cayley graph? (i.e. existence and character of phase transitions, critical behavior) I'm ...
3
votes
0answers
250 views

A general Lipschtiz potential can be specified by a Gibbs specification ?

I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$. Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator $$ ...
12
votes
2answers
1k views

How is the Ising model an example of a lattice model as per Kontsevich?

In section 3.2 of Kontsevich's very interesting paper "Notes on motives in finite characteristic,", he gives an axiomatic definition of a "lattice model" attached to a Boltzmann datum (V_1,V_2,R), ...
2
votes
3answers
741 views

Maximal clique intersection graphs

Consider graph $T$ where nodes correspond to maximal cliques of some graph $G$ and two nodes can be connected if corresponding cliques intersect. Clique tree is an example when $T$ is required to be a ...
2
votes
2answers
327 views

On generalisation of Aizenman-Higuchi Theorem

Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i_1,i_2)$ given by $\|i\|=|i_1|+|i_2|$. For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma_x$ ...
11
votes
3answers
526 views

Exponential bounds for the number of lattice animals with a given boundary.

Hi all, I am doing a work in collaboration with other mathematicians about phase transition in the Ising model and we need to know if exponential upper bounds exist for the number of lattice animals ...
2
votes
0answers
270 views

Self-avoiding Walk with next-nearest neighbors

Background I study polymer physics and am doing experiments testing the model outlined in this paper. Basically, the polymers fall into an integer number of pits, and we create a partition function ...
12
votes
2answers
604 views

Is there a percolation threshold in the hard discs model?

Take a random configuration of $n$ non-overlapping discs of radius $r$ in the unit square $[0,1]^2$. (You could think of this as taking $n$ points uniform randomly in $[r,1-r]^2$ and then restricting ...
3
votes
1answer
323 views

Is there an example of Gibbs measure that is not a weak limit of finite volume Gibbs measure ?

Consider the first neighbors Ising model in $\mathbb Z^2$, with the Hamiltonian in the finite volume $\Lambda\subset\mathbb{Z}^2$ given by $$ ...
6
votes
1answer
437 views

For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?

Background: Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set ...
2
votes
0answers
131 views

finding set of tree decompositions to cover all pairs of vertices

I first asked this on cstheory.SE but got no reply. Let $P(X_i=x)$ represent probability that randomly chosen proper $q$-coloring of an $L\times L$ square grid contains color $x$ at position $i$. How ...