All Questions
Tagged with pr.probability mp.mathematical-physics
158 questions
5
votes
1
answer
365
views
power laws emerging from the sandpile model
Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
- 5,063
15
votes
1
answer
660
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 ...
- 8,071
11
votes
1
answer
626
views
Formula for $U(N)$ integration wanted
Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.
What I would like is a formula ...
CommunityBot
- 1
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
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 ...
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 ...
3
votes
0
answers
191
views
Infinite total variation of complex measure in Feynman path integral [closed]
I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...
- 31
4
votes
1
answer
229
views
How are the real-space RG transformations defined?
I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
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1
vote
1
answer
63
views
$P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity
We start a Brownian motion at $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a large enough ball that contains compact sets $A$ and $B$. In other words, the B.M. starts on the exterior of $A$ and $B$.
Then ...
- 19.6k
6
votes
1
answer
353
views
Quaternion Wishart matrices of half-integer dimension?
For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
$$P_\beta(\...
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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 ...
CommunityBot
- 1
11
votes
2
answers
1k
views
How should a mathematician approach the physics literature concerning percolation?
I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...
- 151k
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
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
5
votes
1
answer
697
views
Harmonic Crystal using Random Walk
Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...
- 13.6k
22
votes
3
answers
6k
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 ...
- 846
6
votes
2
answers
912
views
References for a physicist migrating to stochastic processes
I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
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- 1
3
votes
1
answer
134
views
GOE convergence
As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...
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- 1
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
1
vote
1
answer
143
views
Minimum of Random Energy Model (REM) with logarithmically correlated potential
In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...
- 188k
1
vote
2
answers
949
views
How to evaluate the wiener measure of sets?
I would like to understand how the Wiener measure of some simple sets can be evaluated.
I will sketch the construction of Wiener measure I have in mind:
We denote the one point compactification of $\...
- 8,992
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 ...
1
vote
2
answers
415
views
$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$
Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any $\phi\in\{-u+...
- 1,805
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
1
vote
1
answer
711
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 \...
- 188k
5
votes
2
answers
808
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 ...
- 632
8
votes
3
answers
2k
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 ...
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- 1
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
2
votes
1
answer
566
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 - \lambda_i)^2}\...
CommunityBot
- 1
4
votes
1
answer
645
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 ...
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- 1
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 ...
- 12.3k
19
votes
1
answer
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 ...
CommunityBot
- 1
2
votes
1
answer
245
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 ...
- 41
0
votes
0
answers
208
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 ...
- 19.6k
0
votes
1
answer
369
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 ...
- 7,514
3
votes
2
answers
941
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, ...
- 188k
1
vote
1
answer
835
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 ...
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- 1
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
8
votes
1
answer
527
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 - \frac{\...
CommunityBot
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6
votes
3
answers
423
views
Infinite electrical networks and possible connections with LERW
I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
Given a lattice L, we turn it into a ...
- 151
11
votes
1
answer
1k
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 ...
- 188k
5
votes
1
answer
437
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 ...
CommunityBot
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6
votes
1
answer
844
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 $G$-...
- 561
4
votes
3
answers
1k
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.
...
- 141
8
votes
1
answer
993
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 $$L=\frac{1}{2}(-\...
- 1,111
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 ...
CommunityBot
- 1
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
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
13
votes
7
answers
1k
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 ...
- 513
11
votes
1
answer
1k
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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 ...
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