All Questions
Tagged with mp.mathematical-physics pr.probability
51 questions with no upvoted or accepted answers
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
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
18
votes
0
answers
310
views
Profiles of very high dimensional functions
This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...
- 96.4k
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
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
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 ...
- 10.5k
7
votes
0
answers
497
views
Extreme unitary minimal models of conformal field theory
Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...
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 ...
- 23.2k
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
6
votes
0
answers
360
views
What is the status of the Born Rule in axiomatic QM?
While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
- 237
6
votes
0
answers
334
views
Hints on an expository article about Kardar-Parisi-Zhang (KPZ)
It seems the KPZ is the next big thing in mathematical physics and probability. The skeletal idea is probably that while classical averages are in the Gaussian universality class, lots of other ...
- 267
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 ...
- 22.8k
6
votes
0
answers
262
views
Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
6
votes
0
answers
411
views
Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.8k
5
votes
0
answers
139
views
Stationary point processes with arbitrarily slow decorrelation
A point process $P$ (a probability measure on simple, locally finite point configurations $\mathcal{C}$ on $\mathbb{R}$ - I'm restricting to the one-dimensional setting) is stationary when law-...
- 121
5
votes
0
answers
127
views
First return time in an interval for N particles rotating on the circle at constant random speeds
Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
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
4
votes
0
answers
164
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....
- 435
4
votes
0
answers
334
views
Unusual generalization of the law of large numbers
I have seen in physical literature
an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
- 21.8k
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
3
votes
0
answers
342
views
Sum of products of irreducible characters of the symmetric group over a subgroup
When trying to build a dual formulation for lattice gauge theories using Weingarten integration I am getting sums of the kind
$$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} \chi_\mu (\pi \...
3
votes
0
answers
126
views
Other than Brownian motion, when else is it possible to define "normalized weighted infinite dimensional Lebesgue measure"?
In this article Sourav Chatterjee poses the question, how do we define the measure:
$$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$
The $Z$ here is an infinite normalizing ...
user69208
3
votes
0
answers
112
views
Uniqueness results for lattice spin systems (graphs)
Are there any nice uniqueness results for Gibbs-measures on lattice spin systems (graphs) that does not rely on Dobrushin's method?
- 244
3
votes
0
answers
188
views
Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
- 265
3
votes
0
answers
134
views
SOS model - height
Let $\Lambda_n = \{ -n,\ldots,n\}^2$ and let $\{X_i\}_{i\in \Lambda_n}$ be the family of $\mathbb{R}$-valued of random variables with the density proportional to
$\exp(-\sum_{i\sim j} |X_i - X_j|),$
...
- 1,491
3
votes
0
answers
229
views
For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?
Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J \sum_{(...
- 2,252
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
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
2
votes
0
answers
103
views
Is this correct: Inflection points of Euler number graph in Island-Mainland transition correspond to spanning cluster site percolation threshold?
I'm writing with respect to the 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 a PDF ...
user119567
2
votes
0
answers
491
views
Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question
This is a prequel to my question:
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Clearly my ...
- 1,026
2
votes
0
answers
192
views
Diffusion equation on mixing of diffusing particles
I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diffusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...
2
votes
0
answers
1k
views
Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution?
In Science and Hypothesis, chapter XI, The calculus of probabilities, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/...
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
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
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
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
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
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
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
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
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
1
vote
0
answers
84
views
Particle density in phase space normalization under proliferation
Consider $1,..,N$ indistinguishable particles in $\mathbb{R}^2$ and let them evolve according to a brownian motion and proliferation. Let $u: \mathbb{R}_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}_0^+...
1
vote
0
answers
94
views
Reason of the scaling factor $n^{2}$ in Hydrodynamic limits
In some books about hydrodynamic limits, example De Masi and Pressuti, when taking about the transition from micro to macro to get the hydrodynamic limit of some process it is mentioned that in order ...
1
vote
0
answers
1k
views
What is the characteristic functional for Brownian motion on a sphere?
I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
- 111
1
vote
0
answers
215
views
Has this process been studied?
Take a Poisson process on $\mathbb{R}$ with intensity given by Lebesgue measure. Think of this as the measure $d\mu=\sum_{n} \delta(t-\xi_n )dt$ where $\xi_n$ are the points of the process. Now ...
- 1,471
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
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
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
0
answers
42
views
Probability of close approach for multivariate normal variables
The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
- 195
0
votes
0
answers
257
views
Sum over a product of binomial coefficients related to a collision problem
I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:
$$R\left(n,m,j\right)=\sum_{k=0}^{n}...
- 1