Questions tagged [statistical-physics]
The study of physical systems using probabilistic reasoning, especially relating small-scale classical mechanics to large-scale thermodynamics.
2
votes
1answer
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
0answers
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
0answers
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
1answer
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
3answers
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 $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...
6
votes
0answers
156 views
Is this “stretched eigenvector” studied? (If so, what are its properties?)
An eigenvector is defined by
$$
\lambda \mathbf{v} = A\mathbf{v}.\tag{1}
$$
But suppose I change this to
$$
\lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2}
$$
for real $\alpha\ne 1$, where $\mathbf{v}^...
5
votes
1answer
104 views
An extension of the Izergin-Korepin determinant to the eight-vertex model
In the six-vertex model, edges in a square lattice are oriented so that the in-degree of each vertex is exactly two. This gives six types of allowable vertices:
$$\begin{array}{cccccc}
\begin{...
1
vote
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
$$\propto\prod_{i<...
1
vote
0answers
84 views
Show this function is strictly concave
Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq j}\{\Phi[\sqrt{w}(v-u_k)]\}...
0
votes
0answers
90 views
Basic Monte Carlo Integral Approximation
On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let
\begin{equation}
I = \int_D g(\textbf{x})d\textbf{x},
\end{equation}
where $D \subset \...
2
votes
0answers
54 views
TAP expression for entropy [closed]
This paper by Barton and Cocco: http://www.phys.ens.fr/~cocco/Art/articlejstat.pdf
claims on page 17 (Formula (30)) an expression for the "high-temperature" entropy of an Ising model, given its ...
3
votes
0answers
154 views
Hamiltonian on the torus
In discrete models like Ising we have Hamiltonians of the form
$$H(\sigma)=\frac{1}{N}\sum_{i=1}^{N}J_{ij}\sigma_{i}\sigma_{j},$$
where $\sigma_{i}=\pm 1$ , $J_{ij}$ interaction coefficients and N ...
5
votes
3answers
1k views
Ergodic theory: from Dynamics to Gibbs measure
I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure.
I have a compact space $X$, a probability ...
5
votes
0answers
51 views
Full distribution of FPTs in random walks on graphs
There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ...
5
votes
1answer
302 views
Importance of Ornstein's isomorphism theorem
"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows ...
1
vote
0answers
45 views
PDF of points at the intersection of a sphere and hyperboloid in n dimensions
I'm studying a statistical mechanics problem and I have two conserved quantities:
$$ 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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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 ...
