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5
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1answer
92 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), ...
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0answers
56 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 ...
5
votes
1answer
400 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
112 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 ...
2
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1answer
58 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 ...
2
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0answers
55 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
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2answers
128 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
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1answer
92 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 ...
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0answers
20 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 ...
25
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2answers
828 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 ...
22
votes
1answer
571 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 $2$ ...
6
votes
2answers
544 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. ...
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2answers
522 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. ...
8
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1answer
121 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
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2answers
374 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
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0answers
169 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 ...
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0answers
296 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
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0answers
225 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 ...
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0answers
198 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 ...
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0answers
130 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, ...
18
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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 ...
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0answers
547 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 ...
3
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3answers
1k 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 ...
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0answers
152 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
135 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 ...
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0answers
166 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
175 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 ...
3
votes
2answers
7k 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 ...
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0answers
110 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 ...
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0answers
119 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
653 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 ...
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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 ...
8
votes
1answer
271 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 ...
13
votes
1answer
421 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 ...
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1answer
385 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 ...
2
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1answer
319 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 ...
21
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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 ...
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7answers
997 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 ...
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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 ...
6
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2answers
429 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 ...
1
vote
1answer
197 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 ...
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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
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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), ...
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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 ...
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3answers
550 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 ...
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3answers
774 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 ...
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0answers
278 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 ...
16
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3answers
761 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 ...
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1answer
335 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
$$
...
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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 ...
