Questions tagged [statistical-physics]
The study of physical systems using probabilistic reasoning, especially relating small-scale classical mechanics to large-scale thermodynamics.
80
questions with no upvoted or accepted answers
23
votes
0
answers
2k
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 references)...
14
votes
0
answers
561
views
Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces?
The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...
10
votes
0
answers
376
views
Is there a general theory for Wilsonian renormalization?
I know that Wilson's renormalization group is not a theory per se and that there are many ways to implement it in a given system. Also, renormalization group techniques are applied in a large number ...
9
votes
0
answers
520
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)....
8
votes
0
answers
168
views
Roots of a family of polynomials forming shapes
Let $f$ be a smooth and strictly concave function on $[0,1]$, where $f(0)=f(1)=0$.
Let $F_n(x)=\underset{k=0}{\overset{n} \sum } \exp(nf(\frac kn))x^k$.
The roots of $F_n$ seems to form "shapes&...
8
votes
0
answers
170
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 ...
7
votes
0
answers
587
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 ...
6
votes
0
answers
185
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}^...
6
votes
0
answers
364
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 ...
5
votes
0
answers
232
views
$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder
I'd like to share a problem that I have been dealing with for a longer time now.
In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
5
votes
0
answers
65
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 ...
4
votes
0
answers
267
views
Examples of measures that satisfy FKG, but not the FKG lattice condition
Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...
4
votes
0
answers
145
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....
4
votes
0
answers
189
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 ...
4
votes
0
answers
263
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 ...
4
votes
0
answers
298
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, symmetry,...
3
votes
0
answers
105
views
Differential entropy of random Gibbs measure
There is a question I have been wondering about for a while, which I have thus far not been able to resolve. The problem revolves around random Gibbs measures. I am not very well-versed in the more ...
3
votes
0
answers
74
views
Convergence in perturbative renormalization
Consider the following:
$$G(\phi,W) = -\log \int d\mu_{C}(\psi)e^{-W(\phi+\psi)} \tag{1}\label{1}$$
which is very common in QFT. Here $d\mu_{C}$ is a Gaussian measure with covariance $C$. I want to ...
3
votes
0
answers
76
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.
...
3
votes
0
answers
111
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
74
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^{...
3
votes
0
answers
112
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?
3
votes
0
answers
201
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 ...
3
votes
0
answers
227
views
For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?
Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J \sum_{(...
3
votes
0
answers
226
views
What is the underlying graphical calculus of the Interactions-Round-a-Face lattice model?
Background
Let $\mathcal{L}$ be an $m \times n$ square lattice on a torus, and let $\Sigma$ be a finite set. We think of $\Sigma$ as the possible spin values that can be assigned to the points of the ...
2
votes
0
answers
94
views
Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic
Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
2
votes
0
answers
109
views
What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
2
votes
0
answers
180
views
Harish-Chandra–Itzykson–Zuber integral with two terms
We know
$$
\int \mathcal{D}U \exp(\mathrm{Tr}(AUBU^*))
$$
can be computed by Harish-Chandra–Itzykson–Zuber(HCIZ) integral. I am wondering whether it is possible to compute
$$
I=\int \mathcal{D}U \exp(\...
2
votes
0
answers
76
views
Entropy per site of quantum spin chain
It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...
2
votes
0
answers
61
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 ...
2
votes
0
answers
343
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 ...
2
votes
0
answers
154
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 $d$-...
2
votes
0
answers
68
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) \\
\...
2
votes
0
answers
1k
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 $\rho^W_0(s)=\...
2
votes
0
answers
123
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 ...
2
votes
0
answers
261
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
$$
\mathcal{L}...
2
votes
0
answers
316
views
Self-avoiding Walk with next-nearest neighbors
Background
I study polymer physics and am doing experiments testing the model outlined in this paper. Basically, the polymers fall into an integer number of pits, and we create a partition function ...
2
votes
0
answers
143
views
finding set of tree decompositions to cover all pairs of vertices
I first asked this on cstheory.SE but got no reply.
Let $P(X_i=x)$ represent probability that randomly chosen proper $q$-coloring of an $L\times L$ square grid contains color $x$ at position $i$. How ...
1
vote
0
answers
57
views
Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?
Consider the following non-convex function
$$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$
where $A$ is a symmetric, diagonal-free matrix whose non-diagonal element are $\pm 1$. In other words,...
1
vote
0
answers
136
views
Recommendation to understand mean field theorem
I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
1
vote
0
answers
85
views
Mixing for a gas of hard spheres
The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
1
vote
0
answers
55
views
Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
1
vote
0
answers
125
views
Large-deviation inequalities for a class of simple random multivariate polynomials
Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...
1
vote
0
answers
82
views
Large deviation inequalities for number of coupon types collected by a coupon collector with fixed budget
In the generalized Coupon Collector's Problem, there are $N$ types of coupon, and for any $i \in [N] := \{1,2,\ldots,N\}$, $p_i \ge 0$ is the probability of obtaining a type-i coupon on any trial. ...
1
vote
0
answers
112
views
Mathematical justification for the use of an energy shell in the microcanonical ensemble
I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics.
Consider $\Lambda$ to be ...
1
vote
0
answers
40
views
Langevin dynamics or stochastic gradient flow for grand canonical ensemble
We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity.
Is there any dynamic corresponding to the grand ...
1
vote
0
answers
186
views
Special function: Pulse peak modified with a power term
PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
1
vote
0
answers
135
views
Width of the critical window in a random graph
In an Erdős–Rényi random graph $G(n,p)$, the giant component emerges with thresholding function $p(n) = c/n$, where $c>1$.
When $c=1$, and $\lambda \in \mathbb{R}$, we can write or "...
1
vote
0
answers
147
views
physical interpretation of ruelle probablity cascades (SK model)
Background: the Parisi formula gives an exact expression for the free energy of the SK model. The formula (at least the upper-bound) can be derived by looking at the free energy, and then replacing ...
1
vote
0
answers
85
views
How is the quasipotential in Freidlin-Wentzell theory of large deviations affected by $C^1$ transformations?
I have a 3D differential equation I'm interested in studying the potential landscape using quasipotentials, described in depth in this paper. I need to calculate the potential landscape several times, ...