Questions tagged [statistical-physics]

The study of physical systems using probabilistic reasoning, especially relating small-scale classical mechanics to large-scale thermodynamics.

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Reformulation - Construction of thermodynamic limit for GFF

I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
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5 votes
1 answer
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A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
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41 votes
10 answers
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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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1 answer
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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 ...
Adrien Hardy's user avatar
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1 answer
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map from 6-vertex model to domino tiling

I am trying to find a correspondence between 6-vertex model and an Aztec Diamond tiling. Here are the building blocks of the 8-vertex model: There seems to be more than one correspondence. I found ...
john mangual's user avatar
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7 votes
1 answer
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A counterexample for the Mean Ergodic Theorem in $L_\infty$

The so-called Mean Ergodic Theorem goes back to von Neumann for Hilbert spaces. Later on, versions of this result in reflexive Banach spaces have also appeared (see, e.g., the book by Krengel, Ergodic ...
Antonio J. Urena's user avatar
1 vote
1 answer
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Measure, volume and cardinality on Minlos' book on statistical physics

The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...
MathMath's user avatar
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23 votes
0 answers
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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 references)...
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22 votes
2 answers
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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. ...
asv's user avatar
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14 votes
2 answers
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How are Clifford algebras and spinors used to study the Ising model?

I've heard Clifford algebras and spinors are useful tools to study the Ising model, but I've never find any good discussion on this matter. Also, as far as I know, in the original solutions of the 2-D ...
IamWill's user avatar
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11 votes
3 answers
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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 ...
Marc Kjerland's user avatar
8 votes
2 answers
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Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?

During these first months in my PhD, I realized how my computational problems can be drastically reduced to one single problem: Find an efficient way to sample from a Gibbs measure. Let me ...
duccio's user avatar
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1 answer
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Regularizing divergent sums over lattices

Vertex-models and nearest-neighbor models with translation-invariant local energy functions on an infinite lattice have the troublesome feature that the sum of the local energies diverges. A standard ...
James Propp's user avatar
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6 votes
2 answers
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Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sinks. Why?

The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various ...
TheSimpliFire's user avatar
6 votes
2 answers
475 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), n=2,3,\...
Slava Krushkal's user avatar
5 votes
1 answer
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Vorticial ground states for the O(2) rotor model

Is there a sensible notion of a ground state for the classical $O(2)$ rotor model "frustrated at infinity by a single unit of counterclockwise vorticity"? Here is a picture of the kind of thing I mean,...
James Propp's user avatar
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4 votes
2 answers
260 views

Grand-canonical Gibbs measure for continuous systems

Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
IamWill's user avatar
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4 votes
2 answers
549 views

A relation between the second moment of a distribution and one of its particular probability

I had recently posted a question here: To prove a relation involving a probability distribution The relations quoted in the above question are used extensively in fluid mechanics and many other fields,...
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3 votes
1 answer
215 views

Mathematical meaning for the (continuous) Sine-Gordon transformation

I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be ...
IamWill's user avatar
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2 votes
1 answer
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Singular values of sparse random real-valued matrix

I was wondering if anyone knew of any results regarding the limiting distribution of singular values for sparse random real-valued matrices? Specifically, let $X$ be an $N\times M$ matrix with real-...
user19346's user avatar
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2 answers
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Imprecise Definition of a $\sigma$-algebra

I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
IamWill's user avatar
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1 vote
1 answer
210 views

To prove a relation involving a probability distribution

I'm reading a book and have encountered a relation which seems to me to be impossible to prove, I would like to be sure if this is the case. The author gives a probability function as $$p_n = \frac{e^...
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How to calculate the exact probability $p$ at which maxima occurs in the curves in an infinite system?

I'm writing with respect to this paper: Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice Here's a ...
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