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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 ...
5 votes
1 answer
382 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 ...
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 ...
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}...
1 vote
2 answers
489 views

The limiting behavior of geometric random walk

I would like to know what the asymptotic limiting behavior is for the following random walk on $\mathbb Z^d$. By Donsker's invariance principle, I suspect that its behavior is diffusive, i.e., the ...
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 ...
4 votes
3 answers
644 views

Traceless GUE : Four Centered Fermions

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel $$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
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 ...
6 votes
2 answers
5k views

Can I relate the L1 norm of a function to its Fourier expansion?

I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
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, ...
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. ...
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 ...
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 $...
1 vote
1 answer
505 views

Gaussian measures on non-separable spaces

Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
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 ...
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}{\...
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=...
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}\...
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,...
6 votes
0 answers
398 views

semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int ...
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 ...
7 votes
0 answers
579 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
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, ...
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 ($h_0(x) = 1$, $h_1(x) = x$...
35 votes
5 answers
11k views

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
5 votes
1 answer
945 views

Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?

In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
12 votes
3 answers
1k views

Correlations in last-passage percolation

Consider the last passage percolation model on $\mathbb{Z}^2$ with, say, geometric weights on each edge. By a landmark result of Johansson (http://arxiv.org/abs/math/9903134), we know that if $T_n(\...
74 votes
16 answers
8k views

Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
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 ...
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 ...
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-...
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 $\...
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)...
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-...
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 ...
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, ...
7 votes
0 answers
627 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 ...
21 votes
2 answers
2k views

Uncertainty principle and Cramer-Rao bound - is there relation?

Just out of curiosity. The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound. Saying that we cannot measure something with certain accuracy. However looking closer ...
10 votes
2 answers
2k views

When is a space of measures a measurable space?

Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
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 ...
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 ...
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,...
2 votes
1 answer
161 views

Expected value of global functions in renormalization group

This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance ...
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 ...
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 ...
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 ...
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 ...
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: ...
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, ...
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 ...