Skip to main content

All Questions

Filter by
Sorted by
Tagged with
5 votes
1 answer
383 views

Rigorous statistical mechanics: difficulty of realistic models

Soft question: I am a mathematician self-learning statistical mechanics. The (mathematical) literature is concentrated on lattice models like the Ising model and the lattice-gas model. I understand ...
Plemath's user avatar
  • 312
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
1 vote
0 answers
48 views

Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators

Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation: $$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
user avatar
0 votes
0 answers
59 views

Convergence of Liouville correlation functions

A key object in Liouville conformal field theory is the random Liouville measure $M$ defined heuristically as $M(d^2x) = :e^{2bX(x)}: d^2x$, where $X$ is a Gaussian free field and $:e^{2bX}:$ denotes ...
user avatar
6 votes
0 answers
163 views

Wick ordering, probability vs physics

Consider a collection of creation $a^\dagger$and annihilation operators $a$. In physics one defines Wick ordering (also known as normal ordering) as a prescription to place all creation operators ...
CBBAM's user avatar
  • 721
2 votes
0 answers
115 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. ...
Plemath's user avatar
  • 312
2 votes
1 answer
158 views

Definition of average $\langle \langle \cdot \rangle \rangle$

I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in ...
JustWannaKnow's user avatar
2 votes
1 answer
403 views

Fokker Planck equation in the Stratonovich approach

I'm a physics master student and I have difficulties understanding how to derive the Fokker Planck equation from the Stratonovich SDE. With the Ito SDE it is simple since the noise is independent of $...
user515206's user avatar
20 votes
0 answers
3k views
+200

What does a product of many Gaussian matrices converge to?

Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$. Is ...
Yaroslav Bulatov's user avatar
0 votes
0 answers
112 views

Additivity of purity of random matrix products

Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as $$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...
Yaroslav Bulatov's user avatar
6 votes
1 answer
274 views

Spectrum asymptotics for a product of $k$ random matrices?

How does the spectrum of a product of $k$ random matrices behave around 0? In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
Yaroslav Bulatov's user avatar
10 votes
1 answer
1k views

Proving the Replica Trick works

The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit $$ \log(Z) = \lim_{n\to 0}\...
Felix Benning's user avatar
2 votes
0 answers
129 views

Large deviation principle for product of iid bounded symmetric random variables

Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,...
dohmatob's user avatar
  • 6,853
1 vote
0 answers
129 views

Concentration of a combinatorial sum

Let $X=(x_1,\ldots,x_p)$ be an $p \times n$ random matrix with iid entries from $\{\pm 1\}$, distributed so that $\mathbb P(x_{ij} = 1) \equiv 1/2$, where $x_i=(x_{i1},\ldots,x_{in})$. Let $y$ be a ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
228 views

When is a stationary measure of a Markov chain "exponentially localized"?

Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes. Some intuition can gained by thinking about a diffusion process, ...
Piyush Grover's user avatar
2 votes
1 answer
173 views

Random variables with density distributions given by squared Hermite polynomials

I was wondering whether anything is known on the following: Let $h_k (x)= (-1)^k e^{x^2/2} \frac {d^k}{dx^k} \, e^{-x^2/2}$, $k \geq 0$, be the classical Hermite polynomials ($h_0(x) = 1$, $h_1(x) = x$...
Sunia Cortez's user avatar
6 votes
2 answers
481 views

Infinite clusters for loopless percolation

I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is ...
PeaBrane's user avatar
  • 213
3 votes
1 answer
155 views

What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?

The Tracy–Widom distributions admit many interpretations. One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...
LeechLattice's user avatar
  • 9,501
1 vote
0 answers
164 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 ...
DJA's user avatar
  • 435
2 votes
1 answer
194 views

Kramers' escape problem: statistical physics vs. Large deviations

I'm almost not at all knowledgable in either Freidlin-Wentzel theory or Kramers' escape problem as it is known in the physics community, so please excuse some of my naivety. One can use Freidlin-...
Stefan Perko's user avatar
1 vote
0 answers
65 views

Intuition behind bound of second moment of Greens function by fractional moment

Consider the Hilbert space $ \mathcal{H} = l^2(\mathbb{Z}^d)$ for some dimension $d$ with basis given by the basisvectors $\{ \vert {x} \rangle \}_{x \in \mathbb{Z}^d} $. Let $A$ be an either self-...
Frederik Ravn Klausen's user avatar
6 votes
2 answers
1k views

In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?

The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit. We begin with a Hilbert space $\...
Mehmet Coen's user avatar
10 votes
1 answer
337 views

What are the predictive implications of conditional non-commutative probability?

To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$. In this context a state $S$ is a positive semi-definite ...
Mehmet Coen's user avatar
0 votes
1 answer
133 views

How to demonstrate a correlation inequality? [closed]

If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$. The correlation between Z, Y is greater than between X, ...
Mac Zhang's user avatar
2 votes
1 answer
232 views

An inequality for a "generalised random energy model"

Let, for all $i, j$, $Z_{i,j}$ be a standard normal, chosen iid. For each $n\geq 1, k\geq2$, define the Hamiltonian $H_{n,k}: [k]^n \to \mathbb{R}$ by $$(j_1,j_2,\ldots,j_n) \mapsto \sum_{i=1}^n Z_{i, ...
Vilhelm Agdur's user avatar
1 vote
0 answers
91 views

A random process with conserved momentum: 'particle decay'?

Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...
Leo Moos's user avatar
  • 5,048
2 votes
1 answer
187 views

Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix

Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
dohmatob's user avatar
  • 6,853
4 votes
2 answers
564 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,...
user avatar
1 vote
0 answers
93 views

Does Anderson localisation occur if the potential are equal in pairs?

Consider the Anderson model given by the Hamiltonian $H \in B(l^2( \mathbb{Z}^d)) $ defined by $H = - \Delta + V$ where the potential $V$ acts on a unit vector $ \vert x \rangle  \in l^2( \mathbb{Z}^d)...
Frederik Ravn Klausen's user avatar
3 votes
0 answers
204 views

What should I study to approach the frontier of integrable probability research?

In terms of math, I know measure theory, measure theory based probability, differentiable manifolds, galois theory, some algebraic topology, and some representation theory. I have only physics 101 ...
kid111's user avatar
  • 151
2 votes
1 answer
394 views

Is there a Hilbert space approach to commutative probability theory on locally compact spaces?

I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum ...
Andrew NC's user avatar
  • 2,071
10 votes
2 answers
2k views

Why does Riesz's Representation Theorem apply in quantum mechanics?

$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$. It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
Andrew NC's user avatar
  • 2,071
4 votes
1 answer
243 views

Does there exist a scale invariant random packing of circles in the plane?

I want to construct a scale invariant random packing of the plane with circles. Here is a way to construct a rotationally invariant, but not scale invariant random packing of the plane with circles: ...
Frederik Ravn Klausen's user avatar
0 votes
0 answers
83 views

Random walk in random enviroment

I am looking for a classical analogue of localization for quantum walks. First, I draw for each point in $x \in \mathbb{Z}^2$ (with some distribution) the numbers $u_x,d_x,l_x,r_x$ such that $u_x+d_x+...
Frederik Ravn Klausen's user avatar
0 votes
1 answer
143 views

Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube

The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$ \begin{equation} \label{one} \int_0^1 \...
Paul B. Slater's user avatar
1 vote
0 answers
90 views

Proving that a model exhibits either a first or second order phase transition

Motivating example: Take the (wired) random cluster model $\phi^1_{p,q}$ with parameter $q$ (see http://arxiv.org/abs/1707.00520 for an introduction). It is now known on $\mathbb{Z}^2$ that it has a ...
Frederik Ravn Klausen's user avatar
4 votes
1 answer
237 views

What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure?

This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations: (1) $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, ...
MathMath's user avatar
  • 1,305
11 votes
2 answers
914 views

How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian

Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding. In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous ...
truebaran's user avatar
  • 9,330
4 votes
0 answers
321 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 $ ...
Frederik Ravn Klausen's user avatar
6 votes
1 answer
1k views

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 ...
JustWannaKnow's user avatar
3 votes
3 answers
501 views

Identity on convolution with Gaussian measure

I've came across an identity once (I don't remember where) concerning convolutions of Gaussian measures. If I'm not mistaken, this identity was \begin{eqnarray} (\mu_{C}*f)(y) = \exp\bigg{[}\frac{1}{...
JustWannaKnow's user avatar
6 votes
1 answer
708 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 ...
JustWannaKnow's user avatar
5 votes
2 answers
393 views

Connections between two constructions of infinite dimensional Gaussian measures

Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} ...
JustWannaKnow's user avatar
11 votes
2 answers
353 views

Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
neverevernever's user avatar
5 votes
1 answer
439 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: \...
JustWannaKnow's user avatar
1 vote
0 answers
114 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-...
JustWannaKnow's user avatar
8 votes
0 answers
195 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 ...
Frederik Ravn Klausen's user avatar
1 vote
0 answers
223 views

L2 norm of the diagonal entries of a random rotation of a fixed matrix?

Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
Sitan Chen's user avatar
2 votes
1 answer
238 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}^{|\...
JustWannaKnow's user avatar
6 votes
2 answers
905 views

Gaussian measure on function spaces

I'm reading this classic work and I'd like to get deeper inside some of its techniques. In particular, the authors state: "We construct a Gaussian measure $d\mu_{0}(\phi)$ on a measure space of ...
JustWannaKnow's user avatar