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The latest articles from Electronic Communications in Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Tue, 14 Feb 2017 04:00 ESTTue, 14 Feb 2017 04:00 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The Intrinsic geometry of some random manifolds
http://projecteuclid.org/euclid.ecp/1483585770
<strong>Sunder Ram Krishnan</strong>, <strong>Jonathan E. Taylor</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 22, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.
</p>projecteuclid.org/euclid.ecp/1483585770_20170214040037Tue, 14 Feb 2017 04:00 ESTTail asymptotics of maximums on trees in the critical casehttps://projecteuclid.org/euclid.ecp/1532678499<strong>Mariusz Maślanka</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We consider the endogenous solution to the stochastic recursion \[ X\,{\buildrel \mathit {d}\over =}\,\bigvee _{i=1}^{N}{A_iX_i}\vee{B} ,\] where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in{\mathbb {N}} }$ are random nonnegative numbers and $X_i$ are independent copies of $X$, independent also of $N$, $B$, $\{A_i\}_{i\in \mathbb{N} }$. The properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E} \left [\sum _{i=1}^{N}{A_i^s}\right ]$. Recently, Jelenković and Olvera-Cravioto assuming, inter alia, $m(s)<1$ for some $s$, proved that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e. \[\mathbb{P} [R>t]\sim{Ct^{-\alpha }} \] for some $\alpha >0$ and $C>0$. In this paper we prove an analogous result when $m(s)=1$ has unique solution $\alpha >0$ and $m(s)>1$ for all $s\not =\alpha $.
</p>projecteuclid.org/euclid.ecp/1532678499_20181221220228Fri, 21 Dec 2018 22:02 ESTNon-triviality of the vacancy phase transition for the Boolean modelhttps://projecteuclid.org/euclid.ecp/1533002443<strong>Mathew D. Penrose</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.
</p>projecteuclid.org/euclid.ecp/1533002443_20181221220228Fri, 21 Dec 2018 22:02 ESTCritical radius and supremum of random spherical harmonics (II)https://projecteuclid.org/euclid.ecp/1535767261<strong>Renjie Feng</strong>, <strong>Xingcheng Xu</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 11 pp..</p><p><strong>Abstract:</strong><br/>
We continue the study, begun in [6], of the critical radius of embeddings, via deterministic spherical harmonics, of fixed dimensional spheres into higher dimensional ones, along with the associated problem of the distribution of the suprema of random spherical harmonics. Whereas [6] concentrated on spherical harmonics of a common degree, here we extend the results to mixed degrees, en passant improving on the lower bounds on critical radii that we found previously.
</p>projecteuclid.org/euclid.ecp/1535767261_20181221220228Fri, 21 Dec 2018 22:02 ESTConvergence of maximum bisection ratio of sparse random graphshttps://projecteuclid.org/euclid.ecp/1535767262<strong>Brice Huang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider sequences of large sparse random graphs whose degree distribution approaches a limit with finite mean. This model includes both the random regular graphs and the Erdös-Renyi graphs of constant average degree. We prove that the maximum bisection ratio of such a graph sequence converges almost surely to a deterministic limit. We extend this result to so-called 2-spin spin glasses in the paramagnetic to ferromagnetic regime. Our work generalizes the graph interpolation method to some non-additive graph parameters.
</p>projecteuclid.org/euclid.ecp/1535767262_20181221220228Fri, 21 Dec 2018 22:02 ESTProjections of spherical Brownian motionhttps://projecteuclid.org/euclid.ecp/1535767263<strong>Aleksandar Mijatović</strong>, <strong>Veno Mramor</strong>, <strong>Gerónimo Uribe Bravo</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.
</p>projecteuclid.org/euclid.ecp/1535767263_20181221220228Fri, 21 Dec 2018 22:02 ESTFluctuations for block spin Ising modelshttps://projecteuclid.org/euclid.ecp/1535767264<strong>Matthias Löwe</strong>, <strong>Kristina Schubert</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet et al. in [2]. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we prove a non-standard CLT for the magnetization.
</p>projecteuclid.org/euclid.ecp/1535767264_20181221220228Fri, 21 Dec 2018 22:02 ESTNon-convergence of proportions of types in a preferential attachment graph with three co-existing typeshttps://projecteuclid.org/euclid.ecp/1535767265<strong>John Haslegrave</strong>, <strong>Jonathan Jordan</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We consider the preferential attachment model with multiple vertex types introduced by Antunović, Mossel and Rácz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunović, Mossel and Rácz. We also consider another family of examples where we show that the conjecture does hold.
</p>projecteuclid.org/euclid.ecp/1535767265_20181221220228Fri, 21 Dec 2018 22:02 ESTFractional Brownian motion with zero Hurst parameter: a rough volatility viewpointhttps://projecteuclid.org/euclid.ecp/1536718014<strong>Eyal Neuman</strong>, <strong>Mathieu Rosenbaum</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.
</p>projecteuclid.org/euclid.ecp/1536718014_20181221220228Fri, 21 Dec 2018 22:02 ESTCoalescing random walk on unimodular graphshttps://projecteuclid.org/euclid.ecp/1536804170<strong>Eric Foxall</strong>, <strong>Tom Hutchcroft</strong>, <strong>Matthew Junge</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.
</p>projecteuclid.org/euclid.ecp/1536804170_20181221220228Fri, 21 Dec 2018 22:02 ESTExistence of an unbounded vacant set for subcritical continuum percolationhttps://projecteuclid.org/euclid.ecp/1536977436<strong>Daniel Ahlberg</strong>, <strong>Vincent Tassion</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
We consider the Poisson Boolean percolation model in $\mathbb{R} ^2$, where the radius of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R} ^d$, for any $d\ge 2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.
</p>projecteuclid.org/euclid.ecp/1536977436_20181221220228Fri, 21 Dec 2018 22:02 ESTA Brownian optimal switching problem under incomplete informationhttps://projecteuclid.org/euclid.ecp/1537840941<strong>Marcus Olofsson</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study an incomplete information optimal switching problem in which the manager only has access to noisy observations of the underlying Brownian motion $\{W_t\}_{t \geq 0}$. The manager can, at a fixed cost, switch between having the production facility open or closed and must find the optimal management strategy using only the noisy observations. Using the theory of linear stochastic filtering, we reduce the incomplete information problem to a full information problem, show that the value function is non-decreasing with the amount of information available, and that the value function of the incomplete information problem converges to the value function of the corresponding full information problem as the noise in the observed process tends to $0$. Our approach is deterministic and relies on the PDE-representation of the value function.
</p>projecteuclid.org/euclid.ecp/1537840941_20181221220228Fri, 21 Dec 2018 22:02 ESTA note on tail triviality for determinantal point processeshttps://projecteuclid.org/euclid.ecp/1539763345<strong>Russell Lyons</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 3 pp..</p><p><strong>Abstract:</strong><br/>
We give a very short proof that determinantal point processes have a trivial tail $\sigma $-field. This conjecture of the author has been proved by Osada and Osada as well as by Bufetov, Qiu, and Shamov. The former set of authors relied on the earlier result of the present author that the conjecture held in the discrete case, as does the present short proof.
</p>projecteuclid.org/euclid.ecp/1539763345_20181221220228Fri, 21 Dec 2018 22:02 ESTApproximation of a generalized continuous-state branching process with interactionhttps://projecteuclid.org/euclid.ecp/1539763346<strong>Ibrahima Dramé</strong>, <strong>Étienne Pardoux</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 14 pp..</p><p><strong>Abstract:</strong><br/>
In this work, we consider a continuous–time branching process with interaction where the birth and death rates are non linear functions of the population size. We prove that after a proper renormalization our model converges to a generalized continuous state branching process solution of the SDE \[\begin{aligned} Z_t^x=& x + \int _{0}^{t} f(Z_r^x) dr + \sqrt{2c} \int _{0}^{t} \int _{0}^{Z_{r}^x }W(dr,du) + \int _{0}^{t}\int _{0}^{1}\int _{0}^{Z_{r^-}^x}z \ \overline{M} (ds, dz, du)\\ &+ \int _{0}^{t}\int _{1}^{\infty }\int _{0}^{Z_{r^-}^x}z \ M(ds, dz, du), \end{aligned} \] where $W$ is a space-time white noise on $(0,\infty )^2$ and $\overline{M} (ds, dz, du)= M(ds, dz, du)- ds \mu (dz) du$, with $M$ being a Poisson random measure on $(0,\infty )^3$ independent of $W,$ with mean measure $ds\mu (dz)du$, where $(1\wedge z^2)\mu (dz)$ is a finite measure on $(0, \infty )$.
</p>projecteuclid.org/euclid.ecp/1539763346_20181221220228Fri, 21 Dec 2018 22:02 ESTSquared Bessel processes of positive and negative dimension embedded in Brownian local timeshttps://projecteuclid.org/euclid.ecp/1539763347<strong>Jim Pitman</strong>, <strong>Matthias Winkel</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
The Ray–Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell $ of a reflecting Brownian motion $|B|$ by a multiple $\mu $ of its local time process $\ell $ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison–Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.
</p>projecteuclid.org/euclid.ecp/1539763347_20181221220228Fri, 21 Dec 2018 22:02 ESTA large deviation principle for the Erdős–Rényi uniform random graphhttps://projecteuclid.org/euclid.ecp/1540346603<strong>Amir Dembo</strong>, <strong>Eyal Lubetzky</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
Starting with the large deviation principle (LDP) for the Erdős–Rényi binomial random graph ${\mathcal G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph ${\mathcal G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in ${\mathcal G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.
</p>projecteuclid.org/euclid.ecp/1540346603_20181221220228Fri, 21 Dec 2018 22:02 ESTBlock size in Geometric($p$)-biased permutationshttps://projecteuclid.org/euclid.ecp/1540346604<strong>Irina Cristali</strong>, <strong>Vinit Ranjan</strong>, <strong>Jake Steinberg</strong>, <strong>Erin Beckman</strong>, <strong>Rick Durrett</strong>, <strong>Matthew Junge</strong>, <strong>James Nolen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 10 pp..</p><p><strong>Abstract:</strong><br/>
Fix a probability distribution $\mathbf p = (p_1, p_2, \ldots )$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty )$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.
</p>projecteuclid.org/euclid.ecp/1540346604_20181221220228Fri, 21 Dec 2018 22:02 ESTA sharp symmetrized form of Talagrand’s transport-entropy inequality for the Gaussian measurehttps://projecteuclid.org/euclid.ecp/1540346605<strong>Max Fathi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
This note presents a sharp transport-entropy inequality that improves on Talagrand’s inequality for the Gaussian measure, arising as a dual formulation of the functional Santaló inequality. We also discuss some extensions and connections with concentration of measure.
</p>projecteuclid.org/euclid.ecp/1540346605_20181221220228Fri, 21 Dec 2018 22:02 ESTA renewal theorem and supremum of a perturbed random walkhttps://projecteuclid.org/euclid.ecp/1540346606<strong>Ewa Damek</strong>, <strong>Bartosz Kołodziejek</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest.
We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.
</p>projecteuclid.org/euclid.ecp/1540346606_20181221220228Fri, 21 Dec 2018 22:02 ESTOn pathwise quadratic variation for càdlàg functionshttps://projecteuclid.org/euclid.ecp/1542942174<strong>Henry Chiu</strong>, <strong>Rama Cont</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
We revisit Föllmer’s concept of quadratic variation of a càdlàg function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of càdlàg processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition which implies the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.
</p>projecteuclid.org/euclid.ecp/1542942174_20181221220228Fri, 21 Dec 2018 22:02 ESTHigh points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal processhttps://projecteuclid.org/euclid.ecp/1542942175<strong>Constantin Glenz</strong>, <strong>Nicola Kistler</strong>, <strong>Marius A. Schmidt</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
It has been proved by Bovier & Hartung [ Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean’s martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean’s martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean’s martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, ‘McKean-like martingales’ are naturally expected to arise in all models belonging to the BBM-universality class.
</p>projecteuclid.org/euclid.ecp/1542942175_20181221220228Fri, 21 Dec 2018 22:02 ESTA functional limit theorem for the profile of random recursive treeshttps://projecteuclid.org/euclid.ecp/1542942176<strong>Alexander Iksanov</strong>, <strong>Zakhar Kabluchko</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots , X_{[n^t]}(k))_{t\geq 0}$, for each $k\in \mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment. Let $Y_k(t)$ be the number of the $k$th generation individuals born at times $\leq t$ in this process. Then, it is shown that the appropriately centered and normalized vector-valued process $(Y_{1}(st),\ldots , Y_k(st))_{t\geq 0}$ converges weakly, as $s\to \infty $, to the same limiting Gaussian process as above.
</p>projecteuclid.org/euclid.ecp/1542942176_20181221220228Fri, 21 Dec 2018 22:02 ESTRigidity of the $\operatorname{Sine} _{\beta }$ processhttps://projecteuclid.org/euclid.ecp/1545102490<strong>Chhaibi Reda</strong>, <strong>Joseph Najnudel</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 8 pp..</p><p><strong>Abstract:</strong><br/>
We show that the $\operatorname{Sine} _{\beta }$ point process, defined as the scaling limit of the Circular Beta Ensemble when the dimension goes to infinity, and generalizing the determinantal sine-kernel process, is rigid in the sense of Ghosh and Peres: the number of points in a given bounded Borel set $B$ is almost surely equal to a measurable function of the position of the points outside $B$.
</p>projecteuclid.org/euclid.ecp/1545102490_20181221220228Fri, 21 Dec 2018 22:02 ESTExtremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measureshttps://projecteuclid.org/euclid.ecp/1545102491<strong>Codina Cotar</strong>, <strong>Benedikt Jahnel</strong>, <strong>Christof Külske</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 12 pp..</p><p><strong>Abstract:</strong><br/>
The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.
</p>projecteuclid.org/euclid.ecp/1545102491_20181221220228Fri, 21 Dec 2018 22:02 ESTShifted critical threshold in the loop $ \boldsymbol{O(n)} $ model at arbitrarily small $n$https://projecteuclid.org/euclid.ecp/1545102492<strong>Lorenzo Taggi</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 9 pp..</p><p><strong>Abstract:</strong><br/>
In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to \[\lambda ^{ \# \mbox{ edges} } n^{ \# \mbox{ loops} },\] where $\lambda , n \in [0, \infty )$. Let $\mu $ be the connective constant of the lattice and, for any $n \in [0, \infty )$, let $\lambda _c(n)$ be the largest value of $\lambda $ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $\lambda _c(n) =1/\mu $ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n \geq 0$, $\lambda _c(n) \geq 1/\mu $. In this note we prove that, \[ \lambda _c(n) > 1/\mu \, \, \, \, \forall n >0, \] \[\lambda _c(n) \geq 1/\mu \, + \, c_0 \, n \, + \, O(n^2) \, \, \mbox{ as $n \rightarrow 0$,} \] on $\mathbb{Z} ^d$, with $d \geq 2$, and on the hexagonal lattice, where $c_0>0$. This means that, when $n$ is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk.
</p>projecteuclid.org/euclid.ecp/1545102492_20181221220228Fri, 21 Dec 2018 22:02 ESTOn the supremum of products of symmetric stable processeshttps://projecteuclid.org/euclid.ecp/1545102493<strong>Christophe Profeta</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study the asymptotics, for small and large values, of the supremum of a product of symmetric stable processes. We show in particular that the lower tail exponent remains the same as for only one process, possibly up to some logarithmic terms. The proof relies on a path construction of stable bridges using last sign changes.
</p>projecteuclid.org/euclid.ecp/1545102493_20181221220228Fri, 21 Dec 2018 22:02 ESTThe genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selectionhttps://projecteuclid.org/euclid.ecp/1545102494<strong>Aser Cortines</strong>, <strong>Bastien Mallein</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 13 pp..</p><p><strong>Abstract:</strong><br/>
We study the genealogy of an exactly solvable population model with $N$ particles on the real line, which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around $a$ times its current position, where $a>0$ is a parameter of the model. Then, the $N$ rightmost newborn children are selected to form the next generation. We show that the genealogy of the process converges toward a Beta coalescent as $N \to \infty $. The process we consider can be seen as a toy model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein–Uhlenbeck processes. The parameter $a$ is akin to the pulling strength of the Ornstein–Uhlenbeck process.
</p>projecteuclid.org/euclid.ecp/1545102494_20181221220228Fri, 21 Dec 2018 22:02 ESTA spectral decomposition for the block counting process and the fixation line of the beta(3,1)-coalescenthttps://projecteuclid.org/euclid.ecp/1545447666<strong>Martin Möhle</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 23, 15 pp..</p><p><strong>Abstract:</strong><br/>
A spectral decomposition for the generator of the block counting process of the $\beta (3,1)$-coalescent is provided. This decomposition is strongly related to Riordan matrices and particular Fuss–Catalan numbers. The result is applied to obtain formulas for the distribution function and the moments of the absorption time of the $\beta (3,1)$-coalescent restricted to a sample of size $n$. We also provide the analog spectral decomposition for the fixation line of the $\beta (3,1)$-coalescent. The main tools in the proofs are generating functions and Siegmund duality. Generalizations to the $\beta (a,1)$-coalescent with parameter $a\in (0,\infty )$ are discussed leading to fractional differential or integral equations.
</p>projecteuclid.org/euclid.ecp/1545447666_20181221220228Fri, 21 Dec 2018 22:02 ESTPropagation of chaos for a balls into bins modelhttps://projecteuclid.org/euclid.ecp/1546571102<strong>Nicoletta Cancrini</strong>, <strong>Gustavo Posta</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
Consider a ﬁnite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This ﬁnite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable ( chaotic ) set of initial states, as $L\to +\infty $, the numbers of balls in each bin become independent from the rest of the system i.e. we have propagation of chaos . We furthermore study some equilibrium properties of the limiting nonlinear process .
</p>projecteuclid.org/euclid.ecp/1546571102_20190103220507Thu, 03 Jan 2019 22:05 ESTLimit theorems for the tagged particle in exclusion processes on regular treeshttps://projecteuclid.org/euclid.ecp/1548299047<strong>Dayue Chen</strong>, <strong>Peng Chen</strong>, <strong>Nina Gantert</strong>, <strong>Dominik Schmid</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process “seen from the tagged particle” has an ergodic invariant measure.
</p>projecteuclid.org/euclid.ecp/1548299047_20190123220414Wed, 23 Jan 2019 22:04 ESTScaling of the Sasamoto-Spohn model in equilibriumhttps://projecteuclid.org/euclid.ecp/1548817627<strong>Milton Jara</strong>, <strong>Gregorio R. Moreno Flores</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of [9] and does not use any spectral gap argument.
</p>projecteuclid.org/euclid.ecp/1548817627_20190129220717Tue, 29 Jan 2019 22:07 ESTContinuity and growth of free multiplicative convolution semigroupshttps://projecteuclid.org/euclid.ecp/1548817631<strong>Xiaoxue Deng</strong>, <strong>Ping Zhong</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
Let $\mu $ be a compactly supported probability measure on the positive half-line and let $\mu ^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu ^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.
</p>projecteuclid.org/euclid.ecp/1548817631_20190129220717Tue, 29 Jan 2019 22:07 ESTHeat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductanceshttps://projecteuclid.org/euclid.ecp/1549357292<strong>Sebastian Andres</strong>, <strong>Jean-Dominique Deuschel</strong>, <strong>Martin Slowik</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 17 pp..</p><p><strong>Abstract:</strong><br/>
We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results in [3] to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.
</p>projecteuclid.org/euclid.ecp/1549357292_20190205040139Tue, 05 Feb 2019 04:01 ESTErratum: Nonconventional random matrix productshttps://projecteuclid.org/euclid.ecp/1549530018<strong>Yuri Kifer</strong>, <strong>Sasha Sodin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 1 pp..</p><p><strong>Abstract:</strong><br/>
The proof of Theorem 2.3 in our paper [3] is fully justified only under the additional assumption $q_i(n)=a_in+b_i,\, i=1,...,\ell $.
</p>projecteuclid.org/euclid.ecp/1549530018_20190207040023Thu, 07 Feb 2019 04:00 ESTOn the tails of the limiting QuickSort densityhttps://projecteuclid.org/euclid.ecp/1550113298<strong>James Allen Fill</strong>, <strong>Wei-Chun Hung</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density $f$ that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function $F$. Furthermore, we obtain similar bounds on absolute values of derivatives of $f$ of each order.
</p>projecteuclid.org/euclid.ecp/1550113298_20190213220146Wed, 13 Feb 2019 22:01 ESTDean-Kawasaki dynamics: ill-posedness vs. trivialityhttps://projecteuclid.org/euclid.ecp/1550113299<strong>Vitalii Konarovskyi</strong>, <strong>Tobias Lehmann</strong>, <strong>Max-K. von Renesse</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 9 pp..</p><p><strong>Abstract:</strong><br/>
We prove that the Dean-Kawasaki SPDE admits a solution only in integer parameter regimes, in which case the solution is given in terms of a system of non-interacting particles.
</p>projecteuclid.org/euclid.ecp/1550113299_20190213220146Wed, 13 Feb 2019 22:01 ESTAlmost sure limit theorems on Wiener chaos: the non-central casehttps://projecteuclid.org/euclid.ecp/1550199821<strong>Ehsan Azmoodeh</strong>, <strong>Ivan Nourdin</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].
</p>projecteuclid.org/euclid.ecp/1550199821_20190214220349Thu, 14 Feb 2019 22:03 ESTCritical percolation and the incipient infinite cluster on Galton-Watson treeshttps://projecteuclid.org/euclid.ecp/1550480494<strong>Marcus Michelen</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 13 pp..</p><p><strong>Abstract:</strong><br/>
We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.
</p>projecteuclid.org/euclid.ecp/1550480494_20190218040140Mon, 18 Feb 2019 04:01 ESTExpectation of the largest bet size in the Labouchere systemhttps://projecteuclid.org/euclid.ecp/1550826036<strong>Yanjun Han</strong>, <strong>Guanyang Wang</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 10 pp..</p><p><strong>Abstract:</strong><br/>
For the Labouchere system with winning probability $p$ at each coup, we prove that the expectation of the largest bet size under any initial list is finite if $p>\frac{1} {2}$, and is infinite if $p\le \frac{1} {2}$, solving the open conjecture in [6]. The same result holds for a general family of betting systems, and the proof builds upon a recursive representation of the optimal betting system in the larger family.
</p>projecteuclid.org/euclid.ecp/1550826036_20190222040046Fri, 22 Feb 2019 04:00 ESTConcentration for Coulomb gases on compact manifoldshttps://projecteuclid.org/euclid.ecp/1553133701<strong>David García-Zelada</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 18 pp..</p><p><strong>Abstract:</strong><br/>
We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafaï, Hardy and Maïda on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.
</p>projecteuclid.org/euclid.ecp/1553133701_20190320220157Wed, 20 Mar 2019 22:01 EDTClosed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processeshttps://projecteuclid.org/euclid.ecp/1553133702<strong>Arturo Valdivia</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process. The problem is approached in two ways. On the one hand, we translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials. On the other hand, for Hazard processes driven by Lévy processes, we use Malliavin calculus in order to express the aforementioned distributions in an appealing recursive manner. We outline the potential application of these results in credit risk.
</p>projecteuclid.org/euclid.ecp/1553133702_20190320220157Wed, 20 Mar 2019 22:01 EDTA Hoeffding inequality for Markov chainshttps://projecteuclid.org/euclid.ecp/1553133703<strong>Shravas Rao</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove deviation bounds for the random variable $\sum _{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty }$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves upon previously known bounds in that the dependence is on $\sqrt{a_1^2+\cdots +a_n^2} $ rather than $\max _{i}\{a_i\}\sqrt{n} .$ We also prove deviation bounds for certain types of sums of vector–valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten $\infty $-norm of a random matrix whose entries are obtained from a Markov chain.
</p>projecteuclid.org/euclid.ecp/1553133703_20190320220157Wed, 20 Mar 2019 22:01 EDTA spectral decomposition for a simple mutation modelhttps://projecteuclid.org/euclid.ecp/1553133704<strong>Martin Möhle</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 14 pp..</p><p><strong>Abstract:</strong><br/>
We consider a population of $N$ individuals. Each individual has a type belonging to some at most countable type space $K$. At each time step each individual of type $k\in K$ mutates to type $l\in K$ independently of the other individuals with probability $m_{k,l}$. It is shown that the associated empirical measure process is Markovian. For the two-type case $K=\{0,1\}$ we derive an explicit spectral decomposition for the transition matrix $P$ of the Markov chain $Y=(Y_n)_{n\ge 0}$, where $Y_n$ denotes the number of individuals of type $1$ at time $n$. The result in particular shows that $P$ has eigenvalues $(1-m_{0,1}-m_{1,0})^i$, $i\in \{0,\ldots ,N\}$. Applications to mean first passage times are provided.
</p>projecteuclid.org/euclid.ecp/1553133704_20190320220157Wed, 20 Mar 2019 22:01 EDTError bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector modelshttps://projecteuclid.org/euclid.ecp/1553220032<strong>Lê Vǎn Thành</strong>, <strong>Nguyen Ngoc Tu</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, 165 (2016), no. 6, 1114–1140]. The proof is based on Stein’s method for exchangeable pairs.
</p>projecteuclid.org/euclid.ecp/1553220032_20190321220109Thu, 21 Mar 2019 22:01 EDTConditions for the finiteness of the moments of the volume of level setshttps://projecteuclid.org/euclid.ecp/1553220033<strong>D. Armentano</strong>, <strong>J-M. Azaïs</strong>, <strong>D. Ginsbourger</strong>, <strong>J.R. León</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 8 pp..</p><p><strong>Abstract:</strong><br/>
Let $X(t)$ be a Gaussian random field $\mathbb R^d\to \mathbb R$. Using the notion of $(d-1)$- integral geometric measures , we establish a relation between (a) the volume of level sets, and (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.
</p>projecteuclid.org/euclid.ecp/1553220033_20190321220109Thu, 21 Mar 2019 22:01 EDTCritical Liouville measure as a limit of subcritical measureshttps://projecteuclid.org/euclid.ecp/1553306557<strong>Juhan Aru</strong>, <strong>Ellen Powell</strong>, <strong>Avelio Sepúlveda</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 16 pp..</p><p><strong>Abstract:</strong><br/>
We study how the Gaussian multiplicative chaos (GMC) measures $\mu ^\gamma $ corresponding to the 2D Gaussian free field change when $\gamma $ approaches the critical parameter $2$. In particular, we show that as $\gamma \to 2^{-}$, $(2-\gamma )^{-1}\mu ^\gamma $ converges in probability to $2\mu '$, where $\mu '$ is the critical GMC measure.
</p>projecteuclid.org/euclid.ecp/1553306557_20190322220259Fri, 22 Mar 2019 22:02 EDTSubsequential tightness of the maximum of two dimensional Ginzburg-Landau fieldshttps://projecteuclid.org/euclid.ecp/1553306561<strong>Wei Wu</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
We prove the subsequential tightness of centered maxima of two-dimensional Ginzburg-Landau fields with bounded elliptic contrast.
</p>projecteuclid.org/euclid.ecp/1553306561_20190322220259Fri, 22 Mar 2019 22:02 EDTOn coupling and “vacant set level set” percolationhttps://projecteuclid.org/euclid.ecp/1554170646<strong>Alain-Sol Sznitman</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
In this note we discuss “vacant set level set” percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of ${\mathbb Z}^d$, $d \ge 3$.
</p>projecteuclid.org/euclid.ecp/1554170646_20190401220412Mon, 01 Apr 2019 22:04 EDTA note on transportation cost inequalities for diffusions with reflectionshttps://projecteuclid.org/euclid.ecp/1554429763<strong>Soumik Pal</strong>, <strong>Andrey Sarantsev</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusion processes with suitable drift and diffusion coefficients. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.
</p>projecteuclid.org/euclid.ecp/1554429763_20190404220300Thu, 04 Apr 2019 22:03 EDTSimultaneous boundary hitting by coupled reflected Brownian motionshttps://projecteuclid.org/euclid.ecp/1555034600<strong>Krzysztof Burdzy</strong>. <p><strong>Source: </strong>Electronic Communications in Probability, Volume 24, 12 pp..</p><p><strong>Abstract:</strong><br/>
Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.
</p>projecteuclid.org/euclid.ecp/1555034600_20190411220330Thu, 11 Apr 2019 22:03 EDT