Electronic Journal of Probability
http://ejp.ejpecp.org/
<p><strong> </strong></p>The <strong>Electronic Journal of Probability</strong> (EJP) publishes full-length research articles in probability theory. Short papers, those less than 12 pages, should be submitted first to its sister journal, the <a href="http://ecp.ejpecp.org/" target="_blank">Electronic Communications in Probability</a> (ECP). EJP and ECP share the same editorial board, but with different Editors in Chief.<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-USThe 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.ledoux@math.univ-toulouse.fr (Michel Ledoux (Chief Editor))ejpecp@chafai.net (Djalil Chafaï)Thu, 02 Jan 2014 01:40:36 -0800OJS 2.3.6.0http://blogs.law.harvard.edu/tech/rss60Rank deficiency in sparse random GF$[2]$ matrices
http://ejp.ejpecp.org/article/view/2458
Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $\mathcal{N}(n,m)$ denote the number of left null vectors in $\{0,1\}^m$ for $M$ (including the zero vector), where addition is mod 2. We take $n, m \to \infty$, with $m/n \to \alpha > 0$, while the weight distribution converges weakly to that of a random variable $W$ on $\{3, 4, 5, \ldots\}$. Identifying $M$ with a hypergraph on $n$ vertices, we define the 2-core of $M$ as the terminal state of an iterative algorithm that deletes every row incident to a column of degree 1.<br /><br />We identify two thresholds $\alpha^*$ and $\underline{\alpha\mkern-4mu}\mkern4mu$, and describe them analytically in terms of the distribution of $W$. Threshold $\alpha^*$ marks the infimum of values of $\alpha$ at which $n^{-1} \log{\mathbb{E}[\mathcal{N} (n,m)}]$ converges to a positive limit, while $\underline{\alpha\mkern-4mu}\mkern4mu$ marks the infimum of values of $\alpha$ at which there is a 2-core of non-negligible size compared to $n$ having more rows than non-empty columns. We have $1/2 \leq \alpha^* \leq \underline{\alpha\mkern-4mu}\mkern4mu \leq 1$, and typically these inequalities are strict; for example when $W = 3$ almost surely, $\alpha^* \approx 0.8895$ and $\underline{\alpha\mkern-4mu}\mkern4mu \approx 0.9179$. The threshold of values of $\alpha$ for which $\mathcal{N}(n,m) \geq 2$ in probability lies in $[\alpha^*,\underline{\alpha\mkern-4mu}\mkern4mu]$ and is conjectured to equal $\underline{\alpha\mkern-4mu}\mkern4mu$. The random row-weight setting gives rise to interesting new phenomena not present in the case of non-random $W$ that has been the focus of previous work.Richard W. R. Darling, Mathew D. Penrose, Andrew R. Wade, Sandy L. Zabellhttp://ejp.ejpecp.org/article/view/2458Sun, 14 Sep 2014 00:17:33 -0700Random partitions in statistical mechanics
http://ejp.ejpecp.org/article/view/3244
We consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a "chain of Chinese restaurants" stochastic process. We obtain results for the distribution of the size of the largest component.Nicholas M. Ercolani, Sabine Jansen, Daniel Ueltschihttp://ejp.ejpecp.org/article/view/3244Tue, 09 Sep 2014 22:16:53 -0700Free infinite divisibility for beta distributions and related ones
http://ejp.ejpecp.org/article/view/3448
We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, many of ultraspherical and Student t-distributions have free divisibility indicator 1.Takahiro Hasebehttp://ejp.ejpecp.org/article/view/3448Thu, 04 Sep 2014 22:53:03 -0700Height representation of XOR-Ising loops via bipartite dimers
http://ejp.ejpecp.org/article/view/2449
The XOR-Ising model on a graph consists of random spin configurations on vertices of the graph obtained by taking the product at each vertex of the spins of two independent Ising models. In this paper, we explicitly relate loop configurations of the XOR-Ising model and those of a dimer model living on a decorated, bipartite version of the Ising graph. This result is proved for graphs embedded in compact surfaces of genus $g$. Using this fact, we then prove that XOR-Ising loops have the same law as level lines of the height function of this bipartite dimer model. At criticality, the height function is known to converge weakly in distribution to $\frac{1}{\sqrt{\pi}}$ a Gaussian free field. As a consequence, results of this paper shed a light on the occurrence of the Gaussian free field in the XOR-Ising model. In particular, they provide a step forward in the solution of Wilson's conjecture, stating that the scaling limit of XOR-Ising loops are level lines of the Gaussian free field.Cédric Boutillier, Béatrice de Tilièrehttp://ejp.ejpecp.org/article/view/2449Thu, 04 Sep 2014 22:36:13 -0700Percolation on uniform infinite planar maps
http://ejp.ejpecp.org/article/view/2675
We construct the uniform infinite planar map (UIPM), obtained as the $n \to \infty$ local limit of planar maps with $n$ edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are $p^{\textrm{bond}}_c=1/2$ and $p^{\textrm{site}}_c=2/3$ respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is $p^{\textrm{bond}}_c=1/3$.Laurent Ménard, Pierre Nolinhttp://ejp.ejpecp.org/article/view/2675Tue, 02 Sep 2014 23:11:12 -0700