User max - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:42:04Z http://mathoverflow.net/feeds/user/12105 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94307/reference-request-cover-time-for-simple-random-walk-on-nxn-torus/94309#94309 Answer by Max for Reference Request: Cover time for simple random walk on nxn torus Max 2012-04-17T17:50:17Z 2012-04-17T17:50:17Z <p>Markov Chains and Mixing Times by Levin, Peres and Wilmer. Section 11.3.2. </p> <p><a href="http://research.microsoft.com/en-us/um/people/peres/markovmixing.pdf" rel="nofollow">http://research.microsoft.com/en-us/um/people/peres/markovmixing.pdf</a></p> <p>The expected cover time is of order $n^2(\log n)^2$.</p> http://mathoverflow.net/questions/93787/affect-of-noise-on-random-variable-separation/93788#93788 Answer by Max for Affect of noise on Random variable separation Max 2012-04-11T17:45:30Z 2012-04-11T17:45:30Z <p>It seems that you need more assumptions on $X$ and $Y$. Otherwise, take both to be constant and $X = x &lt; Y = y$, so that $P_1 = 0$. Then it is very easy to find examples such that $P_2 > 0$.</p> <p>However, this does not mean that the two random variables are made more separable...</p> http://mathoverflow.net/questions/56356/for-examples-in-probability/56357#56357 Answer by Max for for examples in probability Max 2011-02-23T02:42:35Z 2011-02-23T02:42:35Z <p>Consider independent random variables $X_n, Y_n, n\in\mathbb N$, such that $\mathbb P(X_n = 1) = 1-\mathbb P(X_n = 0) = 1/n$, $\mathbb P(Y_n = n) = 1-\mathbb P(Y_n = 0) = 1/n$. Set $Z_n = X_nY_n$. Then, $Z_n$ is uniformly integrable and $Z_n\to 0$ as $n\to\infty$ almost surely, by Borel-Cantelli Lemma. However, set ${\cal A} = \sigma(X_n:n\in\mathbb N)$. Then $\mathbb E(Z_n\mid{\cal A}) = X_n\mathbb E(Y_n\mid {\cal A}) = X_n$, which converges to 0 in $L^1$ but not almost surely.</p> http://mathoverflow.net/questions/4172/where-does-a-math-person-go-to-learn-statistical-mechanics/56352#56352 Answer by Max for Where does a math person go to learn statistical mechanics? Max 2011-02-23T01:51:19Z 2011-02-23T01:51:19Z <p><a href="http://www.amazon.com/Statistical-Mechanics-Disordered-Systems-Probabilistic/dp/0521849918" rel="nofollow">Statistical Mechanics of Disordered Systems, a Mathematical Perspective</a>, by Anton Bovier. Part I provides a brief introduction to statistical mechanics, while the rest two parts focus on disordered systems. To my understanding, the author treats the problems from a mathematical point of view. </p> http://mathoverflow.net/questions/94307/reference-request-cover-time-for-simple-random-walk-on-nxn-torus/94309#94309 Comment by Max Max 2012-04-18T00:49:17Z 2012-04-18T00:49:17Z For the case of $n\times n$ grid, I don't know the exact rate but I can provide an upper bound on the cover time. By the same Matthews method (Theorem 11.2 of the reference), the cover time is bounded from above by the (largest) hitting time multiplied by $2\log n$. The largest hitting time in this case is of order $n^2\log n$, which can be calculated through effective resistance and commute time identity (Propositions 9.16 and 10.6). Therefore, the cover time in the grid case can be bounded by $O(n^2(\log n)^2)$ too. http://mathoverflow.net/questions/94307/reference-request-cover-time-for-simple-random-walk-on-nxn-torus/94309#94309 Comment by Max Max 2012-04-18T00:40:46Z 2012-04-18T00:40:46Z Actually, if you look at the notes on p. 152, for your original problem, the expected cover time $E(\tau_{cov}) \sim \frac4\pi n^2(\log n)^2$.