All Questions
16 questions
19
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 ...
- 2,242
6
votes
1
answer
271
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=...
- 2,242
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
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
3
votes
1
answer
833
views
Sampling from a particular multivariate probability distribution
Given $3$ real variables $x_1, x_2, x_3 \equiv \bf{x}$, consider their probability density function (PDF)
\begin{equation}
P({\bf x}) = C \, p(x_1) \cdots p(x_3) \exp[f({\bf x})],
\end{equation}
where ...
- 343
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 ...
- 161
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?
- 6,962
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 ...
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, ...
- 33
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 ...
- 8,512
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
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 ...
- 8,512
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 ...
0
votes
1
answer
915
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 \frac{\...
- 33
8
votes
3
answers
2k
views
randomness in nature [closed]
What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
- 307
5
votes
3
answers
601
views
Monte Carlo simulations
I was wondering what were the models of statistical physics that are still considered difficult/slow to simulate (exactly, or approximately) with the current technology of Monte Carlo approaches. I ...
- 2,133