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As Von Neumann said "Nobody really know what entropy is." so it is quite difficult to give a conceptual reason. However I think your calculation appears and can be interpreted in the statistical-mecha …

Let $u\in \mathbb{R}^n$ as $u_i=i^{-1}$. Then $$ \mathbb{E}\big[\langle u Au\rangle \big]=\sum_{i=1}^n i^{-1}-\frac{\epsilon}{2}\sum_{i,j\leq n}\frac{1}{ij}\approx \log n -\frac{\epsilon}{2}(\log n)² …

The "miracle" is just that $T$ is also defined $\mathbb{R}_n[X]\rightarrow \mathbb{R}_n[X]$, a elementary linear operator on finite dimensional space of polynomes. And on this set we have $$T=\begin{ …

I have the following counter example: Let $I=Z_2\times G$ with $G$ a free groupe of rank $d$ and we will take $d$ large. $P_1$ is as follow $$P_1 \begin{cases}(1,w)\rightarrow (0,w):\forall w\in G, …

You should have a look on the book "Product of random matrices and application to Schrodinger operators" (Lacroix, Bougerol) http://fr.booksc.org/book/32773481/48bc02 or the paper of Le Page "Théorème …

You have the very powerful probabilist theorem $$ u(t,x) = \mathbb{E}_x(u(0,B_{t \wedge T})) $$ where $B_t$ is a Brownian process starting at $x$ and $T$ is the stopping time $T=\inf{s : B_s \notin \O …

Case 1: Let $f\geq 0$. We consider the following easier problem :$$ \|f-G_n^*\|_{L^1}\leq \inf \{\|f-G_n\|_{L^1}:a_{i,n}\geq 0, \mu_{i,n}\in \mathbb{R}, \sigma_{i,n}>0 \}. $$ Since $f\geq 0$, we have …

I have a question related to the boltzmann equation and the meaning of the marginals. Let me first introdiuce the model and notation : (see for example https://arxiv.org/abs/1208.5753) We study th …

I would advise you to first look on random walk on trees. I suspect non-amenable group has been treated a lot as well but I can't find a good reference. For example the random walk on the 4 regular t …

You can use Fourier transform (it has been invented for that). Call $\hat{u}(k)$, $k\in \mathbb{R}^d$ the Fourier transform of $u$, for example any function depending only on the absolute value of $|\ …

I am not sure that it is going to work but it seems to me that this is the Legendre transform: $$\|x-h\|^2+\frac{1}{2}f(h)=\|x\|^2-2\Re\langle x,h\rangle+\|h\|^2+\frac{1}{2}f(h) $$ (Where we see $H$ a …

No, you have the simple counter example: $$g_{T}(x)=\frac{1}{T}1_{0\leq x\leq T}. $$ Since $f\in L^1$, there exist $M>0$ with $\|f|_{[-M,M]}\|_{L^1}=\|f\|_{L^1}-\epsilon/N$. Moreover for any $(\beta_i …

What you want looks like very much the "Uniform Uncertainty Principle" in this paper of Candes and Tao https://statweb.stanford.edu/~candes/papers/OptimalRecovery.pdf which is proved for both Bernoul …

I denote $\Delta^{(n)}$ the discrete Laplacien on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) Consider the scaling $ v \rightarrow \phi(t) = \sqrt{n} …

What you want for 2- seems to me similar to a large deviation result (https://en.wikipedia.org/wiki/Large_deviations_theory) of speed $n^2$ for the diagonal term $$\frac{1}{n^2}\log\mathbb{P}(\sum_i …