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Any of these conditions can be waived by permission of the Corresponding Author. ledoux@math.univ-toulouse.fr (Michel Ledoux (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 02 Jan 2014 01:40:36 -0800 OJS http://blogs.law.harvard.edu/tech/rss 60 Malliavin matrix of degenerate SDE and gradient estimate http://ejp.ejpecp.org/article/view/3120 In this article, we prove that the inverse of Malliavin matrix belongs to $L^p(\Omega,\mathbb{P})$ for a class of degenerate stochastic differential equation (SDE). The conditions required are similar to Hörmander's bracket condition, but we  don't need all coefficients of the SDE are smooth. Furthermore, we obtain a locally uniform estimate for the Malliavin matrix and a gradient estimate. We also prove that the semigroup generated by the SDE is strong Feller. These results are illustrated through examples. Zhao Dong, Xuhui Peng http://ejp.ejpecp.org/article/view/3120 Fri, 15 Aug 2014 23:36:57 -0700 The Gaussian free field in interlacing particle systems http://ejp.ejpecp.org/article/view/3732 We show that if an interlacing particle system in a two-dimensional lattice is a determinantal point process, and the correlation kernel can be expressed as a double integral with certain technical assumptions, then the moments of the fluctuations of the height function converge to that of the Gaussian free field. In particular, this shows that a previously studied random surface growth model with a reflecting wall has Gaussian free field fluctuations. Jeffrey Kuan http://ejp.ejpecp.org/article/view/3732 Fri, 15 Aug 2014 06:20:36 -0700 Euclidean partitions optimizing noise stability http://ejp.ejpecp.org/article/view/3083 The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_{i}\}_{i=1}^{k}$ of $\mathbb{R}^{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability.  That is, for $\rho&gt;0$, we maximize$$\sum_{i=1}^{k}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}1_{A_{i}}(x)1_{A_{i}}(x\rho+y\sqrt{1-\rho^{2}})e^{-(x_{1}^{2}+\cdots+x_{n}^{2})/2}e^{-(y_{1}^{2}+\cdots+y_{n}^{2})/2}dxdy.$$Isaksson and Mossel guessed the best partition for this problem and proved some applications of their conjecture. For example, the Standard Simplex Conjecture implies the Plurality is Stablest Conjecture. For $k=3,n\geq2$ and $0&lt;\rho&lt;\rho_{0}(k,n)$, we prove the Standard Simplex Conjecture. The full conjecture has applications to theoretical computer science and to geometric multi-bubble problems (after Isaksson and Mossel). Steven Heilman http://ejp.ejpecp.org/article/view/3083 Fri, 15 Aug 2014 06:05:28 -0700 Regenerative tree growth: structural results and convergence http://ejp.ejpecp.org/article/view/3040 <p>We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n&gt;=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.</p> Jim Pitman, Douglas Rizzolo, Matthias Winkel http://ejp.ejpecp.org/article/view/3040 Fri, 15 Aug 2014 05:50:43 -0700 First critical probability for a problem on random orientations in $G(n,p)$. http://ejp.ejpecp.org/article/view/2725 We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $\{a\to s\}$ (there exists a directed path from $a$ to $s$) and $\{s\to b\}$. We prove that asymptotically the correlation is negative for small $p$, $p&lt;\frac{C_1}n$, where $C_1\approx0.3617$, positive for $\frac{C_1}n&lt;p&lt;\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\frac{C_2}n$, with $C_2\approx7.5$. We conjecture that the correlation  then stays negative for $p$ up to the previously known zero at $\frac12$; for larger $p$ it is positive. Sven Erick Alm, Svante Janson, Svante Linusson http://ejp.ejpecp.org/article/view/2725 Thu, 14 Aug 2014 01:32:23 -0700