All Questions
Tagged with pr.probability mp.mathematical-physics
158 questions
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 ...
- 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 ...
- 109
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}...
user530766
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 ...
user528837
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 ...
- 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. ...
- 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 ...
- 3,515
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 $...
- 21
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 ...
- 2,252
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}{\...
- 2,252
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=...
- 2,252
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}\...
- 385
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,...
- 6,853
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 ...
- 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, ...
- 2,281
2
votes
1
answer
172
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$...
- 23
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 ...
- 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 ...
- 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 ...
- 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-...
- 677
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-...
- 1,054
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 $\...
- 323
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 ...
- 323
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, ...
- 11
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, ...
- 526
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 ...
- 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 ...
- 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,...
user318428
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)...
- 1,054
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 ...
- 151
2
votes
1
answer
393
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 ...
- 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 ...
- 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:
...
- 1,054
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+...
- 1,054
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 \...
- 723
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 ...
- 1,054
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, ...
- 1,305
11
votes
2
answers
913
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
...
- 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 $ ...
- 1,054
6
votes
1
answer
999
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 ...
- 3,515
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}{...
- 3,515
6
votes
1
answer
707
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 ...
- 3,515
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} ...
- 3,515
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}...
- 1,495
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:
\...
- 3,515
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-...
- 3,515
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 ...
- 1,054
1
vote
0
answers
222
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 $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
- 91
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}^{|\...
- 3,515
6
votes
2
answers
904
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 ...
- 3,515