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I wonder where $\log n$ came from. I'm getting just $n$. Can you spot any errors in the argument below? I'll assume that the distance from the origin to each cube vertex is $1$. Claim 1: Let $x$ be …

I believe that the answer heavily depends on the distribution and, thereby, is incomprehensible. Indeed, let us consider the $2\times n$ table (2 rows, n columns). I'll use the random rearrangement ve …

The argument below is not very elegant,but it is, indeed, a standard exercise. Let $g=\max(f-1,0)$. We shall prove that $$ f\log f\le 2g+\frac 2{p-1}g^p\,. $$ The integration and Holder then give the …

Something is strange (about the question) because, due to symmetry, the expectation of $X$ is just $(c,c)$ where $c$ is the expectation of the random variable with the density $p(x)$ proportional to $ …

It is actually more like $e^{-\sqrt n}$. Let's look at the norm of the inverse matrix. The entries are $\pm\prod_{i:i\ne j}\frac 1{z_j-z_i}\sigma_m(z_1,\dots,z_{j-1},z_{j+1},\dots,z_n)$ where $z_k=e^{ …

Warning: This is not a proper answer, just a dump of the thoughts I have had about this problem so far. Also, I'm not an expert in random matrix theory, so some bounds I'll be using may cry for improv …