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Data Scientist @ Explorer Media. Statistics, Geometry and Physics.


Mar
8
awarded  Necromancer
Mar
5
awarded  Popular Question
Jan
22
awarded  Nice Question
Jan
20
revised Understanding zeta function regularization
deleted 10 characters in body
Jan
7
reviewed Approve suggested edit on Real-world applications of mathematics, by arxiv subject area?
Jan
1
comment Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
@oferzeitouni Oops... $SU(n)$ Haar measure is $\Delta^2$ while Zeilberger has $\Delta^{-1}$. I have written an interesting quantity different from Zeilberger's. He does not integrate, he cites Morris identity, basically Selberg integral. What have I written then? Does $\displaystyle \prod_{1 \leq i < j \leq n} \frac{1}{(x_i - x_j)^2}$ have a matrix meaning, or is it easier to use residue calculus directly at this point?
Jan
1
comment Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
@oferzeitouni Browsing through random matrix notes here and here, this might be related to orthogonal matrices in $SO(n)$ rather than unitary $SU(n)$.
Jan
1
comment Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
@oferzeitouni Sharp eye! I took the integral from a paper by D. Zeilberger it says $$ \int_{\mathbb{T}_n} \prod_{i=1}^n \frac{dx_i}{2\pi i x_i} \frac{1}{(1 - x_i)^2} \prod_{1 \leq i < j \leq n} \frac{1}{ x_i - x_j} = \prod_{i=1}^n \frac{1}{i+1} \binom{2i}{i}$$ I am learning if we integrate with respect to Haar measure there would be an extra factor of the Vandermonde determinant. $$\int_{\mathbb{T}_n} \prod_{i=1}^n \frac{dx_i}{2\pi i x_i} \frac{1}{(1 - x_i)^2} \prod_{1 \leq i < j \leq n} \frac{1}{ (x_i - x_j)^{\color{red} 2}} $$
Jan
1
revised Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
tittle is more accurate description of question
Jan
1
revised Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
clarified question as per comments of @QiaochuYuan
Jan
1
comment Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
@QiaochuYuan By resolvent I just mean $(1 - X)^{-1}=1 + X + X^2 + \dots$, let $X = \rho(G)$ where $\rho:G \to \mathrm{GL}(V)$ is a representation. Definitely mis-using the word resolvent here, but can't think of a better word for it.
Dec
31
comment Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
@QiaochuYuan Maybe I got it wrong? I said it was the resolvent of the regular representation. And then you take the determinant of that - which is the product of the eigenvalues. If we set $x = e^{2\pi i t}$ we get the Weyl integration formula, where $$ f(x) = \prod_{i=1}^n \frac{1}{(1 - x_i)^2}$$ So then I asked myself "Which representation is this?"
Dec
31
comment Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
@oferzeitouni asymptotics are okay. i thought it could be tractable since I am only counting the copies of the trivial representation $\mathbf{1}$. The original question was phrased in terms of generating functions with no mention of representation theory representation theory. $$ \int_{\mathbb{T}_n} \prod_{i=1}^n \frac{dx_i}{2\pi i x_i} \frac{1}{(1 - x_i)^2} \prod_{1 \leq i < j \leq n} \frac{1}{ x_i - x_j} $$ Clearly they are integrating along the maximal torus of $U(N)$.
Dec
31
asked Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$
Dec
31
asked dense lattices in high dimensions
Dec
31
awarded  Custodian
Dec
31
reviewed Approve suggested edit on Have any long-suspected irrational numbers turned out to be rational?
Dec
31
reviewed Approve suggested edit on Whitening a random bit sequence
Dec
5
comment efficient arithmetic with (short) Conway games?
Your choice of Haskell - a functional programming language is very interesting considering the recursive nature of Conway games. Nathan Siegel's CGSuite is written in Java. It was last updated in 2011. You may wish to build off his code or start your own.
Dec
4
asked Thurston-Cannon $S^2$-filling curves