# Questions tagged [statistical-physics]

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

142
questions

**4**

votes

**1**answer

103 views

### 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 ...

**1**

vote

**1**answer

70 views

### Convergence of probability measures which (asymptotically) concentrate along a submanifold

Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...

**4**

votes

**1**answer

90 views

### Effective action, partition function and the renormalization group

Mayer expansions and the Hamilton–Jacobi equation by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form:
\...

**5**

votes

**1**answer

113 views

### Renormalization group strategies

Before introducing block spin transformations in chapter four of Random Walks, Critical Phenomena and Triviality in Quantum Field Theory, the authors state the following:
"In this chapter we sketch ...

**1**

vote

**0**answers

93 views

### Spins in classical statistical mechanics

I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-...

**8**

votes

**0**answers

80 views

### What are the tempered Gibbs measures of classical $\phi^4$-theory?

I consider classical $\phi^4$-theory on the lattice. The model is defined in finite volume with Hamiltonian
\begin{align*}
H(\phi) = - \sum_{x \sim y} J_{x,y} \phi_x \phi_y
\end{align*}
and a-priori ...

**1**

vote

**1**answer

140 views

### Thermodynamic limit and Gaussian measures

Let $\Lambda \subset \mathbb{Z}^{d}$ be finite and fixed and consider $\mathbb{R}^{|\Lambda|}$ be the vector space of all sequences $\varphi = (\varphi_{x})_{x\in \Lambda}$. We equip $\mathbb{R}^{|\...

**0**

votes

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62 views

### Sine-Gordon transformation and functional integrals

In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, ...

**4**

votes

**1**answer

79 views

### Identifications between different phase spaces

I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is ...

**1**

vote

**0**answers

100 views

### Measure on a set and its value on $\emptyset$

After my first post here, I have one more doubt which is bothering me. It concerns Minlos's book Introduction to mathematical statistical physics again. To fix the notation, we have $\Lambda \subset \...

**1**

vote

**1**answer

121 views

### 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 ...

**4**

votes

**2**answers

201 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 \{+...

**0**

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**0**answers

54 views

### Reference for kinetic theory on manifolds

I am looking for a reference for kinetic theory on (Riemannian) manifolds. (In particular for mean-field limits for the Vlasov equation in this setting.)
In 'A review of the mean field limits for ...

**2**

votes

**2**answers

266 views

### 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 ...

**3**

votes

**1**answer

164 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 ...

**1**

vote

**1**answer

106 views

### A combinatorial identity on even spanning subgraphs in the Erdös-Renyi random graph with relations to the Ising model

Let $x \in \lbrack 0,1 \rbrack$. Then for any finite graph $G$ consider the Erdös-Renyi random graph where we independently keep each of the edges with probability $x$. Denote the corresponding ...

**3**

votes

**1**answer

207 views

### Examples of particle systems with higher-order collisions

In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...

**8**

votes

**2**answers

628 views

### 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 ...

**0**

votes

**0**answers

96 views

### Computing the partition function via one of the three methods

I am trying to compute the averaged partition function for some system (with very large $N$) and I reach this point:
\begin{equation}
\left < Z\right > =\int \prod_i^N \left (\frac{\...

**3**

votes

**0**answers

59 views

### Using Kac-Rice formula to count average number of sub-regions carved out by $n$ random hyper-planes

Context. This is the first in a set of tiny pieces of a problem I've formulated to help me measure the "complexity" of certain piecewise linear functions. Thanks in advance for your help and patience.
...

**4**

votes

**2**answers

269 views

### Stationary distribution of a Markov process defined on the space of permutations

Let $S$ be the set of $n!$ permutations of the first $n$ integers. Let $p\in(0,1)$. Consider the Markov Process defined on the elements of $S$.
Let $x\in S$. Choose $1\le i <i+1 < n$ uniformly ...

**2**

votes

**1**answer

353 views

### Understanding Finite Size Scaling in Percolation Theory

Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...

**4**

votes

**0**answers

86 views

### List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have?
I am aware of some of the more famous results, e....

**2**

votes

**0**answers

36 views

### Transversal deviation in first passage percolation

Take the lattice $\mathbb{L}^{2}=(\mathbb{Z}^{2},\mathbb{E}^{2})$ with i.i.d. $\text{U}[0,1]$ weights on the edges, and the random variable $D$ giving the maximal transversal deviation of the geodesic ...

**3**

votes

**1**answer

88 views

### Expected size of matchings in a cubic graph

Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...

**5**

votes

**1**answer

202 views

### Switching oriented paths in a graph

Consider an oriented graph (e.g. a finite part of the standard grid with some random orientations).
Each minute the following operation takes place: we choose uniformly randomly an ordered pair $(A,B)...

**2**

votes

**1**answer

117 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

39 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

76 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 ...

**6**

votes

**1**answer

244 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

192 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

170 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

124 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{...

**2**

votes

**1**answer

186 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

78 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

67 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

340 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

142 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

467 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

129 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

943 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 ...

**9**

votes

**0**answers

364 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

176 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

287 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

182 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

100 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

160 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

104 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

62 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^{...

**7**

votes

**1**answer

278 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 ...