Electronic Communications in Probability http://ecp.ejpecp.org/ <p><strong></strong>The <strong>Electronic Communications in Probability</strong> (ECP) publishes short research articles in probability theory. Its sister journal, the <a href="http://ejp.ejpecp.org/" target="_self">Electronic Journal of Probability</a> (EJP), publishes full-length articles in probability theory. Short papers, those less than 12 pages, should be submitted to ECP first. EJP and ECP share the same editorial board, but with different Editors in Chief.</p><p>EJP and ECP are free access official journals of the <a href="http://www.imstat.org/">Institute of Mathematical Statistics</a> (IMS) and the <a href="http://isi.cbs.nl/BS/bshome.htm"> Bernoulli Society</a>. This web site uses the <a href="http://en.wikipedia.org/wiki/Open_Journal_Systems">Open Journal System</a> (OJS) free software developed by the non-profit organization <a href="http://en.wikipedia.org/wiki/Public_Knowledge_Project">Public Knowledge Project</a> (PKP).</p><p>Please consider donating to the <a href="http://www.imstat.org/publications/open.htm" target="_blank">Open Access Fund</a> of the IMS at this <a href="https://secure.imstat.org/secure/orders/donations.asp" target="_blank"><strong>page</strong></a> to keep the journal free.</p> en-US The Electronic Journal of Probability applies the <a href="http://creativecommons.org/licenses/by/2.5/legalcode" target="_blank">Creative Commons Attribution License</a> (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.<br /><br /><strong>Summary of the Creative Commons Attribution License</strong><br /><br />You are free<br /><ul><li> to copy, distribute, display, and perform the work</li><li> to make derivative works</li><li> to make commercial use of the work</li></ul>under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author. ecp@iam.uni-bonn.de (Anton Bovier (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 09 Jan 2014 00:05:12 -0800 OJS 2.3.6.0 http://blogs.law.harvard.edu/tech/rss 60 A maximal inequality for supermartingales http://ecp.ejpecp.org/article/view/3237 A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to$1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming. Bruce Hajek http://ecp.ejpecp.org/article/view/3237 Thu, 14 Aug 2014 00:15:55 -0700 Growing random 3-connected maps or Comment s'enfuir de l'Hexagone http://ecp.ejpecp.org/article/view/3314 We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. As n tends to infinity, the probability that the n'th map in the sequence is 3-connected tends to 2^8/3^6. The sequence of maps has an almost sure limit G, and we show that G is the distributional local limit of large, uniformly random 3-connected graphs. Louigi Addario-Berry http://ecp.ejpecp.org/article/view/3314 Tue, 12 Aug 2014 20:44:38 -0700 On criteria of disconnectedness of $\Lambda$-Fleming-Viot support http://ecp.ejpecp.org/article/view/3208 The totally disconnectedness of support for super Brownian motion in high dimensions is well known. In this paper, we prove that similar results also hold for $\Lambda$-Fleming-Viot process with Brownian spatial motion provided that the associated $\Lambda$-coalescent does not come down from infinity fast enough. Our proof is another application of the lookdown particle representation for $\Lambda$-Fleming-Viot process. We also discuss the disjointness of independent $\Lambda$-Fleming-Viot supports and ranges in high dimensions. The disconnectedness of the $\Lambda$-Fleming-Viot support remains open in certain low dimensions. Xiaowen Zhou http://ecp.ejpecp.org/article/view/3208 Mon, 11 Aug 2014 13:02:31 -0700 Disjoint crossings, positive speed and deviation estimates for first passage percolation http://ecp.ejpecp.org/article/view/3490 Consider bond percolation on the square lattice \(\mathbb{Z}^2\) where each edge is independently open with probability \(p.\) For some positive constants \(p_0 \in (0,1), \epsilon_1\) and \(\epsilon_2,\) the following holds: if \(p &gt; p_0,\) then with probability at least \(1-\frac{\epsilon_1}{n^{4}}\) there are at least \(\frac{\epsilon_2 n}{\log{n}}\) disjoint open left-right crossings in \(B_n := [0,n]^2\) each having length at most \(2n,\) for all \(n \geq 2.\) Using the proof of the above, we obtain positive speed for first passage percolation with independent and identically distributed edge passage times \(\{t(e_i)\}_i\) satisfying \(\mathbb{E}\left(\log{t(e_1)}\right)^+&lt;\infty;\) namely, \(\limsup_n \frac{T_{pl}(0,n)}{n} \leq Q\) a.s. for some  constant \(Q &lt; \infty,\) where \(T_{pl}(0,n)\) denotes the minimum passage time from the point \((0,0)\) to the line \(x=n\) taken over all paths contained in \(B_n.\) Finally, we also obtain deviation corresponding estimates for nonidentical passage times satisfying \(\inf_i\mathbb{P}(t(e_i) = 0) &gt; \frac{1}{2}.\) Ghurumuruhan Ganesan http://ecp.ejpecp.org/article/view/3490 Mon, 11 Aug 2014 12:48:38 -0700 The probability that planar loop-erased random walk uses a given edge http://ecp.ejpecp.org/article/view/2908 We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an $n \times n$ square is used in a loop-erased walk connecting opposite sides is of order $n^{-3/4}$. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants. Gregory Lawler http://ecp.ejpecp.org/article/view/2908 Tue, 05 Aug 2014 06:27:03 -0700