# Questions tagged [statistical-physics]

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

**2**

votes

**1**answer

84 views

### Upper bound on the number of induced subgraphs of the square lattice with all degrees even

An induced subgraph of a graph $G$ is defined by a subset of vertices of $G$ together with all edges in $G$ that connect vertices from the chosen subset.
Consider now an $n\times m$ square lattice. ...

**1**

vote

**0**answers

26 views

### Vertical and horizontal percolation on heterogeneous honeycomb lattice

I have a regular honeycomb lattice where a bond in the unit cell aligns with $(1,0)$; call this the horizontal direction. Each horizontal bond in the lattice is open with probability $p$ and each "...

**0**

votes

**0**answers

57 views

### How to mathematically justify the “sampling” over only $100$ random matrices to estimate percolation thresholds?

As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by Stauffer et al., the variation of spanning cluster percolation probability $\Pi$ in a finite $L < \infty$ square ...

**5**

votes

**1**answer

147 views

### 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 ...

**0**

votes

**3**answers

134 views

### Single quantum particle entropy

Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...

**6**

votes

**0**answers

156 views

### Is this “stretched eigenvector” studied? (If so, what are its properties?)

An eigenvector is defined by
$$
\lambda \mathbf{v} = A\mathbf{v}.\tag{1}
$$
But suppose I change this to
$$
\lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2}
$$
for real $\alpha\ne 1$, where $\mathbf{v}^...

**5**

votes

**1**answer

104 views

### An extension of the Izergin-Korepin determinant to the eight-vertex model

In the six-vertex model, edges in a square lattice are oriented so that the in-degree of each vertex is exactly two. This gives six types of allowable vertices:
$$\begin{array}{cccccc}
\begin{...

**1**

vote

**1**answer

113 views

### Wasserstein distance to the set of Gaussians ; Boltzman dissipation rate

I am interested in the $2$-Wasserstein distance for probabilities over ${\mathbb R}^n$,
$$W_2(\mu,\nu)=\left(\inf\int_{{\mathbb R}^n\times{\mathbb R}^n}|w-v|^2\pi(v,w)\right)^{1/2}$$
where the infimum ...

**0**

votes

**0**answers

77 views

### 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 link to the PDF ...

**0**

votes

**0**answers

63 views

### Not exactly directed percolation

Is the following problem known/well-studies? I'm looking for references or a name that I can look up.
I start with $N$ cell, each one divides into two cells, each one of the new cells either dies ...

**4**

votes

**1**answer

321 views

### Critical Exponents for Island Mainland Transition (Percolation Theory)

I was looking at this paper “Islands in Sea” and “Lakes in Mainland” phases and related transitions simulated on a square lattice on Percolation theory. The concept of phase transition used here seems ...

**4**

votes

**1**answer

115 views

### an application of nth moment of Poisson distribution with stirling number

I was reading the paper on arixv.
I was confused the equation of nth moment of Poisson distribution.
The detail and partial paper as follow:
...
For large N, this connection probability takes ...

**6**

votes

**3**answers

380 views

### What is the link between the Domino Tilings and the Ising Model?

Is there a link between the theory of Domino Tilings and the Ising Model? In the global qualitative sense that physicists use, the answer is "yes". The connections could go like this:
The dimer ...

**0**

votes

**0**answers

79 views

### Expected value of parametrized Gibbs distribution w.r.t another probability distribution

Let $\mu$ be a compactly supported probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ has density. For a random point $x \sim \mu$,
consider ...

**0**

votes

**0**answers

268 views

### What is a self-consistent equation in percolation theory

I was reading papers about percolation theory in which I was confused by the expression "self-consistent equation", for example in Temporal percolation in activity-driven networks. I read some ...

**8**

votes

**0**answers

257 views

### Mysterious relationship between central charges of conformal field theories and the Beraha numbers

Background:
Conformal field theories (CFTs) in two dimensions are partially characterized by a so-called central charge (characterizing the central extension of the Virasoro algebra which defines it)....

**2**

votes

**0**answers

88 views

### Relationship between the Hurst exponent and the alpha parameter

I have a question about the relationship between the Hurst exponent $H$ and the $\alpha$ parameter in the autocorrelation function when long memory is present. As we know in this case the decay of the ...

**12**

votes

**1**answer

257 views

### Chromatic number of the plane and phase transitions of Potts models

There is a simple connection between ground states of antiferromagnetic Potts models and colorings of the plane: if the unit distance graph of the plane ($G=(\mathbb R^2,\{\{x,y\},d_2(x,y)=1\})$) is ...

**4**

votes

**2**answers

129 views

### Functions of correlated random variables

Suppose that $X$ and $Y$ are positive and square-integrable random variables such that $X$ and $Y$ are positively correlated, i.e., $\mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \geq 0$. Let $f: \...

**2**

votes

**0**answers

67 views

### A proof for this equivalent version of the Infrared Bound/Gaussian Domination

I have recently asked this question in Physics Stackexchange, but as there was no success there, a friend pointed out that I might have a better shot here.
Consider the Ising Model in the $d$-...

**3**

votes

**1**answer

136 views

### C^1 fractals in statistical mechanics

It is well-known--even famous--that the Schramm-Loewner curves appear as domain boundaries between phases at second-order critical points like the critical Ising model or percolation in two dimensions....

**3**

votes

**0**answers

96 views

### Conditional expectation with respect to paths of a Markov jump process

I'm having some trouble detangeling how the conditional expectation in equation (2.13) in the article https://arxiv.org/abs/cond-mat/9811220 (Lebowitz, Spohn) is defined.
The context is as follows: ...

**3**

votes

**0**answers

61 views

### Finding analytic expressions for the cumulants of a correlated random variable

I am working with cumulants of a distribution. I have an example of how the second cumulant may be simplified from:
$k_2 = p\alpha^2\left\{\left(\sum a_i\right)^2 - 2\sum_{i<j}a_ia_j\left(1-\rho^{...

**6**

votes

**1**answer

226 views

### 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 ...

**4**

votes

**1**answer

241 views

### Ising model, phase transition

What is the temperature for the phase transition in the triangular-lattice Ising model? and in the hexagonal-lattice Ising model?

**5**

votes

**1**answer

218 views

### rigorous derivation of isoperimetric inequality from ideal gas equation

I'm an undergraduate math student that learned about classical ideal gases and the associated maxwell-boltzmann distribution for particle velocities in a statistical physics course. Now, starting from ...

**5**

votes

**0**answers

309 views

### Lagrangian formulation of the Ising model as a conformal field theory

An important example of conformal field theory is the 2d Ising model, more precisely its scaling limit when the size of the lattice goes to zero. I am not an expert in the field, but this is the only ...

**1**

vote

**3**answers

252 views

### Evaluation of Gaussian density integral

Is there any closed form, asymptotics, and/or approximations for the following integral:
$$f(c) := \frac{1}{\sqrt{2\sigma^2\pi}}\int e^{-x^2/2\sigma^2}\frac{x^2}{1-cx^2}dx,$$ where $\sigma^2$ is real ...

**1**

vote

**1**answer

175 views

### 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-...

**3**

votes

**0**answers

101 views

### Uniqueness results for lattice spin systems (graphs)

Are there any nice uniqueness results for Gibbs-measures on lattice spin systems (graphs) that does not rely on Dobrushin's method?

**7**

votes

**1**answer

312 views

### 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 ...

**5**

votes

**1**answer

241 views

### 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,...

**3**

votes

**0**answers

111 views

### Remaining models conjectured to converge to SLE(6) or CLE(6)

I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic ...

**2**

votes

**0**answers

55 views

### Reference to Semi-Statistical Optimal Control Theory

I don't know what to call what I want to do, so I'll explain and please refer me to texts and papers.
Given a standard control problem,
\begin{align}
\min_{u(t), W} &\int dt\ f(x(t), u(t); W) \\
\...

**0**

votes

**0**answers

47 views

### What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities.
It states: 1.) Given a non-linear, a monotonous ...

**3**

votes

**2**answers

302 views

### Matrix model for “$\beta$-Ginibre” ensembles

A very well known result in random matrix theory is that there exists "nice" (i.e., with independent entries) tridiagonal matrix for the $\beta$-ensembles of random matrix theory
$$\propto\prod_{i<...

**1**

vote

**0**answers

84 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 j}\{\Phi[\sqrt{w}(v-u_k)]\}...

**0**

votes

**0**answers

90 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 \...

**2**

votes

**0**answers

54 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 ...

**3**

votes

**0**answers

154 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

**3**answers

1k 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

**0**answers

51 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 ...

**5**

votes

**1**answer

302 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 ...

**1**

vote

**0**answers

45 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[ a_1^...

**0**

votes

**0**answers

160 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

**2**answers

295 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 ...

**6**

votes

**1**answer

305 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,\...

**2**

votes

**1**answer

146 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

**1**answer

543 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

**1**answer

139 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 ...