0
votes
0answers
190 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: ...
1
vote
1answer
82 views

Stokes-Einstein rotational diffusion and vector orientation time

The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as: $D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi ...
8
votes
3answers
354 views

References request: constructive quantum field theory

I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
1
vote
0answers
196 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 ...
2
votes
1answer
229 views

Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value: $$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - ...
3
votes
0answers
165 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 ...
0
votes
0answers
80 views

Master Equation to Fokker-Planck for a Jump-Diffusion

Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
2
votes
1answer
185 views

Probability measures on $L^p$

Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
1
vote
0answers
107 views

Brownian particles in a box: the probability that a sphere (of some radius) centered on a particle only contains one particle for a duration of time

Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a ...
4
votes
1answer
304 views

Path integrals for stochastic equations

Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example http://arxiv.org/abs/hep-ph/9912209v1 For imaginary time rigorous ...
0
votes
1answer
120 views

How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X

Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
4
votes
0answers
123 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 ...
3
votes
2answers
270 views

Probability distribution for two-state system that depends on residence time

I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
1
vote
1answer
180 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
2answers
213 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 ...
3
votes
0answers
95 views

Lower bounds of laplace transform of characteristic functions

Cross-posted on maths.stackexchange I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function ...
8
votes
1answer
553 views

What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...
0
votes
1answer
278 views

From Brownian Motion to the Heat Equation

Consider a set of N balls that start at the origin. In a given unit of time, $\delta t$, the balls have a probability $p = 0.5$ of jumping a distance $\delta x$ to the right, and the same probability ...
5
votes
0answers
149 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 ...
4
votes
2answers
781 views

Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form $$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix} $$ where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
10
votes
0answers
304 views

Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...
6
votes
2answers
398 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 ...
10
votes
1answer
463 views

Integration over the orthogonal group

Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is: $$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$ This is quite a tricky question: (1) The first thought would go to ...
29
votes
4answers
2k 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 ...
4
votes
1answer
232 views

Stationary, ergodic measures from the structuralist point of view

Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
8
votes
1answer
349 views

A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?

Buried in the physics paper by Nekrasov and Okounkov, a strange identity is proven: $$ \prod_{n > 0} (1 - q^n)^{\mu^2-1} = \sum_{\mathbf{k}} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - ...
6
votes
0answers
236 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
votes
0answers
865 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 ...
4
votes
1answer
470 views

Random geometries

Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all ...
5
votes
2answers
468 views

Blow-up for the quasilinear heat equation $u_t= u \ u_{x x}$ or the related $w_t= \left(w_x e^w\right)_x$

What kind of approaches can be used to study the following quasilinear parabolic pde for a scalar function $u=u(x,t)$ ? $$ u_t= u \ u_{x x} $$ The physical problem where this pde comes from dictates ...
3
votes
0answers
150 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 ...
3
votes
0answers
114 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 $\exp(-\sum_{i\sim j} |X_i - X_j|),$ ...
5
votes
3answers
411 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!} ...
5
votes
1answer
659 views

Path integral and harmonic oscillator

Maybe this is not a research level question. I post it because I heard that the path integral can be rigorous by Brownian motion. But my knowledge of probability is so limited. If ...
9
votes
2answers
926 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 ...
9
votes
3answers
548 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 ...
1
vote
0answers
156 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 ...
10
votes
7answers
961 views

Probabilistic (and other mathematical) methods of physics without the physics?

Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually ...
15
votes
5answers
3k 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?
18
votes
1answer
1k views

Horst Knörrer's Permutation Cancellation Problem

The Problem: The following question of Horst Knörrer is a sort of toy problem coming from mathematical physics. Let $x_1, x_2, \dots, x_n$ and $y_1,y_2,\dots, y_n$ be two sets of real numbers. We ...
10
votes
1answer
955 views

Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?

I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result. The background for this problem comes from the composition of Brownian motion and ...
-1
votes
0answers
217 views

probability distribution for several variables

The Fokker-Planck equation for several variables is : $\frac{\partial W}{\partial t} = L_{FP}W$ where $L_{FP} = -\frac{\partial}{\partial x_i}D_i(\{x\})+\frac{\partial^2}{\partial x_i \partial ...
0
votes
1answer
842 views

Can you interpret this divergent integral?

In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function: $$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 ...
1
vote
0answers
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 ...
4
votes
1answer
621 views

A simple problem in markov chains

I'm trying to understand a 1954 paper of Kubo intitled "Note on the stochastic theory of resonance absorption". The specific problem can be stated mathematically as follows: let $X(t)$ be a random ...
3
votes
0answers
200 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 ...
4
votes
3answers
751 views

Imaginary exponential functional of Brownian motion

Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion: $X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$ where $g$ is a real scale parameter. ...
23
votes
2answers
576 views

What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
6
votes
1answer
432 views

For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?

Background: Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set ...
45
votes
1answer
4k views

The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...