Electronic Journal of Probability Articles (Project Euclid)
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Lévy Classes and Self-Normalization
http://projecteuclid.org/euclid.ejp/1453756464
<strong>Davar Khoshnevisan</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 1, 18 pp..</p><p><strong>Abstract:</strong><br/>
We prove a Chung's law of the iterated logarithm for recurrent linear Markov processes. In order to attain this level of generality, our normalization is random. In particular, when the Markov process in question is a diffusion, we obtain the integral test corresponding to a law of the iterated logarithm due to Knight.
</p>projecteuclid.org/euclid.ejp/1453756464_20160125161428Mon, 25 Jan 2016 16:14 ESTConstruction and Skorohod representation of a fractional $K$-rough pathhttp://projecteuclid.org/euclid.ejp/1498010465<strong>Aurélien Deya</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 40 pp..</p><p><strong>Abstract:</strong><br/>
We go ahead with the study initiated in [7] about a heat-equation model with non-linear perturbation driven by a space-time fractional noise. Using general results from Hairer’s theory of regularity structures, the analysis reduces to the construction of a so-called $K$-rough path (above the noise), a notion we introduce here as a compromise between regularity structures formalism and rough paths theory. The exhibition of such a $K$-rough path at order three allows us to cover the whole roughness domain that extends up to the standard space-time white noise situation. We also provide a representation of this abstract $K$-rough path in terms of Skorohod stochastic integrals.
</p>projecteuclid.org/euclid.ejp/1498010465_20170721220437Fri, 21 Jul 2017 22:04 EDTAsymptotic freeness for rectangular random matrices and large deviations for sample covariance matrices with sub-Gaussian tailshttp://projecteuclid.org/euclid.ejp/1498010466<strong>Benjamin Groux</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 40 pp..</p><p><strong>Abstract:</strong><br/>
We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo’s result for Wigner matrices having the same type of entries [7]. To this aim, we need to establish an asymptotic freeness result for rectangular free convolution, more precisely, we give a bound in the subordination formula for information-plus-noise matrices.
</p>projecteuclid.org/euclid.ejp/1498010466_20170721220437Fri, 21 Jul 2017 22:04 EDTA duality principle in spin glasseshttp://projecteuclid.org/euclid.ejp/1500689052<strong>Antonio Auffinger</strong>, <strong>Wei-Kuo Chen</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 17 pp..</p><p><strong>Abstract:</strong><br/>
We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature parameter, we show that this relation is valid in a large class of disordered systems. In particular, when applied to mean field spin glasses, this duality provides an interpretation of the Parisi formula as an inverted variational principle, establishing a prediction of Guerra [13].
</p>projecteuclid.org/euclid.ejp/1500689052_20170721220437Fri, 21 Jul 2017 22:04 EDTMixing time of the fifteen puzzlehttp://projecteuclid.org/euclid.ejp/1485831705<strong>Ben Morris</strong>, <strong>Anastasia Raymer</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 29 pp..</p><p><strong>Abstract:</strong><br/>
We show that the mixing time for the fifteen puzzle in an $n \times n$ torus is on the order of $n^4 \log n$.
</p>projecteuclid.org/euclid.ejp/1485831705_20170808220101Tue, 08 Aug 2017 22:01 EDTDouble roots of random polynomials with integer coefficientshttp://projecteuclid.org/euclid.ejp/1486090890<strong>Ohad N. Feldheim</strong>, <strong>Arnab Sen</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 23 pp..</p><p><strong>Abstract:</strong><br/>
We consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most $\tfrac 12$, then the probability of the polynomial to have a double root is dominated by the probability that either $0$, $1$, or $-1$ is a double root up to an error of $o(n^{-2})$. We also show that if the support of the coefficients’ distribution excludes $0$, then the double root probability is $O(n^{-2})$. Our result generalizes a similar result of Peled, Sen and Zeitouni [13] for Littlewood polynomials.
</p>projecteuclid.org/euclid.ejp/1486090890_20170808220101Tue, 08 Aug 2017 22:01 EDTUniqueness of critical Gaussian chaoshttp://projecteuclid.org/euclid.ejp/1486090891<strong>Janne Junnila</strong>, <strong>Eero Saksman</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 31 pp..</p><p><strong>Abstract:</strong><br/>
We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.
</p>projecteuclid.org/euclid.ejp/1486090891_20170808220101Tue, 08 Aug 2017 22:01 EDTIntersection and mixing times for reversible chainshttp://projecteuclid.org/euclid.ejp/1486090892<strong>Yuval Peres</strong>, <strong>Thomas Sauerwald</strong>, <strong>Perla Sousi</strong>, <strong>Alexandre Stauffer</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 16 pp..</p><p><strong>Abstract:</strong><br/>
We consider two independent Markov chains on the same finite state space, and study their intersection time , which is the first time that the trajectories of the two chains intersect. We denote by $t_I$ the expectation of the intersection time, maximized over the starting states of the two chains. We show that, for any reversible and lazy chain, the total variation mixing time is $O(t_I)$. When the chain is reversible and transitive, we give an expression for $t_I$ using the eigenvalues of the transition matrix. In this case, we also show that $t_I$ is of order $\sqrt{n \mathbb {E}\!\left [I\right ]} $, where $I$ is the number of intersections of the trajectories of the two chains up to the uniform mixing time, and $n$ is the number of states. For random walks on trees, we show that $t_I$ and the total variation mixing time are of the same order. Finally, for random walks on regular expanders, we show that $t_I$ is of order $\sqrt{n} $.
</p>projecteuclid.org/euclid.ejp/1486090892_20170808220101Tue, 08 Aug 2017 22:01 EDTIntermediate disorder directed polymers and the multi-layer extension of the stochastic heat equationhttp://projecteuclid.org/euclid.ejp/1486090893<strong>Ivan Corwin</strong>, <strong>Mihai Nica</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 49 pp..</p><p><strong>Abstract:</strong><br/>
We consider directed polymer models involving multiple non-intersecting random walks moving through a space-time disordered environment in one spatial dimension. For a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling (in which time and space are scaled diffusively, and the strength of the environment is scaled to zero in a critical manner) the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this paper we prove the analogous result for multiple non-intersecting random walks started and ended grouped together. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O’Connell and Warren.
</p>projecteuclid.org/euclid.ejp/1486090893_20170808220101Tue, 08 Aug 2017 22:01 EDTConditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical casehttp://projecteuclid.org/euclid.ejp/1487127642<strong>Martin Kolb</strong>, <strong>Mladen Savov</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 29 pp..</p><p><strong>Abstract:</strong><br/>
In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. This complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity. We also show that the end point of the process conditioned on survival up to time $t$ rescaled by $\sqrt{t} $ converges in distribution to a non-trivial random variable, as $t$ tends to infinity, which is in fact invariant with respect to the drift $h>0$. We thus prove that it is sub-ballistic and estimate the speed of escape. The latter is in a sharp contrast with discrete models of dimension larger or equal to $2$ when the behaviour at criticality is ballistic, see [7], and even to many one dimensional models which exhibit ballistic behaviour at criticality, see [8].
</p>projecteuclid.org/euclid.ejp/1487127642_20170808220101Tue, 08 Aug 2017 22:01 EDTGrowth-fragmentation processes and bifurcatorshttp://projecteuclid.org/euclid.ejp/1487127643<strong>Quan Shi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 25 pp..</p><p><strong>Abstract:</strong><br/>
Markovian growth-fragmentation processes introduced by Bertoin model a system of growing and splitting cells in which the size of a typical cell evolves as a Markov process $X$ without positive jumps. We find that two growth-fragmentations associated respectively with two processes $X$ and $Y$ (with different laws) may have the same distribution, if $(X,Y)$ is a b ifurcator, roughly speaking, which means that they coincide up to a bifurcation time and then evolve independently. Using this criterion, we deduce that the law of a self-similar growth-fragmentation is determined by its index of self-similarity and a cumulant function $\kappa $.
</p>projecteuclid.org/euclid.ejp/1487127643_20170808220101Tue, 08 Aug 2017 22:01 EDTCritical window for the configuration model: finite third moment degreeshttp://projecteuclid.org/euclid.ejp/1487127644<strong>Souvik Dhara</strong>, <strong>Remco van der Hofstad</strong>, <strong>Johan S.H. van Leeuwaarden</strong>, <strong>Sanchayan Sen</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 33 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.
</p>projecteuclid.org/euclid.ejp/1487127644_20170808220101Tue, 08 Aug 2017 22:01 EDTFunctional central limit theorem for subgraph counting processeshttp://projecteuclid.org/euclid.ejp/1487127645<strong>Takashi Owada</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 38 pp..</p><p><strong>Abstract:</strong><br/>
The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.
</p>projecteuclid.org/euclid.ejp/1487127645_20170808220101Tue, 08 Aug 2017 22:01 EDTLocal limit of the fixed point foresthttp://projecteuclid.org/euclid.ejp/1487127646<strong>Tobias Johnson</strong>, <strong>Anne Schilling</strong>, <strong>Erik Slivken</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 26 pp..</p><p><strong>Abstract:</strong><br/>
Consider the following partial “sorting algorithm” on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is $1$. This process imposes a forest structure on the set of all permutations of size $n$, where the roots are the permutations starting with $1$ and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this “fixed point forest” exhibits a rich structure. In this paper, we consider the fixed point forest in the limit $n\to \infty $ and show using Stein’s method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path from a random permutation to a leaf converges to the geometric distribution with mean $e-1$, and the length of the shortest path converges to the Poisson distribution with mean $1$. In addition, the higher moments are bounded and hence the expectations converge as well.
</p>projecteuclid.org/euclid.ejp/1487127646_20170808220101Tue, 08 Aug 2017 22:01 EDTOn asymptotic behavior of the modified Arratia flowhttp://projecteuclid.org/euclid.ejp/1487386997<strong>Vitalii Konarovskyi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 31 pp..</p><p><strong>Abstract:</strong><br/>
We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [20]. The system is a natural generalization of the coalescing Brownian motions [3, 25]. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptotic behavior of the particles and local properties of the mass distribution at the initial time.
</p>projecteuclid.org/euclid.ejp/1487386997_20170808220101Tue, 08 Aug 2017 22:01 EDTMuttalib–Borodin ensembles in random matrix theory — realisations and correlation functionshttp://projecteuclid.org/euclid.ejp/1498183245<strong>Peter J. Forrester</strong>, <strong>Dong Wang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 43 pp..</p><p><strong>Abstract:</strong><br/>
Muttalib–Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form $\prod _{1 \le j < k \le N}(\lambda _k - \lambda _j) (\lambda _k^\theta - \lambda _j^\theta )$. We study the Laguerre and Jacobi versions of this model — so named by the form of the one-body interaction terms — and show that for $\theta \in \mathbb Z^+$ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general $\theta > 0$, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent result due to Cheliotis, although our derivation is different. We make use of a generalisation of a double contour integral formula for the correlation functions contained in a paper by Adler, van Moerbeke and Wang to analyse the global density (which we also analyse by studying characteristic polynomials), and the hard edge scaled correlation functions. For the global density functional equations for the corresponding resolvents are obtained; solving this gives the moments in terms of Fuss–Catalan numbers (Laguerre case — a known result) and particular binomial coefficients (Jacobi case). For $\theta \in \mathbb Z^+$ the Laguerre and Jacobi cases are closely related to the squared singular values for products of $\theta $ standard Gaussian random matrices, and truncations of unitary matrices, respectively. At the hard edge the double contour integral formulas provide a double contour integral form of the scaled correlation kernel obtained by Borodin in terms of Wright’s Bessel function.
</p>projecteuclid.org/euclid.ejp/1498183245_20170808220101Tue, 08 Aug 2017 22:01 EDTHypoelliptic multiscale Langevin diffusions: large deviations, invariant measures and small mass asymptoticshttp://projecteuclid.org/euclid.ejp/1498809677<strong>Wenqing Hu</strong>, <strong>Konstantinos Spiliopoulos</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 38 pp..</p><p><strong>Abstract:</strong><br/>
We consider a general class of hypoelliptic Langevin diffusions and study two related questions. The first question is large deviations for hypoelliptic multiscale diffusions as the noise and the scale separation parameter go to zero. The second question is small mass asymptotics of (a) the invariant measure corresponding to the hypoelliptic Langevin operator and of (b) related hypoelliptic Poisson equations. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale hypoelliptic diffusion is consistent with the large deviations behavior of its overdamped counterpart. Additionally, we rigorously obtain an asymptotic expansion of the solution to relevant hypoelliptic Poisson equations with respect to the mass parameter, characterizing the order of convergence as the mass parameter goes to zero. The proof of convergence of invariant measures is of independent interest, as it involves an improvement of the hypocoercivity result for the kinetic Fokker-Planck equation. We do not restrict attention to gradient drifts and our proof provides explicit information on the dependence of the bounds of interest in terms of the mass parameter.
</p>projecteuclid.org/euclid.ejp/1498809677_20170808220101Tue, 08 Aug 2017 22:01 EDTTransport-entropy inequalities on locally acting groups of permutationshttp://projecteuclid.org/euclid.ejp/1502244025<strong>Paul-Marie Samson</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 33 pp..</p><p><strong>Abstract:</strong><br/>
Following Talagrand’s concentration results for permutations picked uniformly at random from a symmetric group [27], Luczak and McDiarmid have generalized it to more general groups of permutations which act suitably ‘locally’. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentration properties are consequences of these inequalities. The results are also generalised to a larger class of measures including Ewens distributions of arbitrary parameter $\theta $ on the symmetric group. By projection, we derive transport-entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law. These results are new examples, in discrete setting, of weak transport-entropy inequalities introduced in [7], that contribute to a better understanding of the concentration properties of measures on permutations groups. One typical application is deviation bounds for the so-called configuration functions, such as the number of cycles of given lenght in the cycle decomposition of a random permutation.
</p>projecteuclid.org/euclid.ejp/1502244025_20170808220101Tue, 08 Aug 2017 22:01 EDTInversion, duality and Doob $h$-transforms for self-similar Markov processeshttp://projecteuclid.org/euclid.ejp/1487386998<strong>Larbi Alili</strong>, <strong>Loïc Chaumont</strong>, <strong>Piotr Graczyk</strong>, <strong>Tomasz Żak</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 18 pp..</p><p><strong>Abstract:</strong><br/>
We show that any $\mathbb{R} ^d\setminus \{0\}$-valued self-similar Markov process $X$, with index $\alpha >0$ can be represented as a path transformation of some Markov additive process (MAP) $(\theta ,\xi )$ in $S_{d-1}\times \mathbb{R} $. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X} $ the self-similar Markov process which is obtained from the MAP $(\theta ,-\xi )$ through this extended Lamperti transformation. Then we prove that $\widehat{X} $ is in weak duality with $X$, with respect to the measure $\pi (x/\|x\|)\|x\|^{\alpha -d}dx$, if and only if $(\theta ,\xi )$ is reversible with respect to the measure $\newcommand{\ed } {\stackrel{(d)} {=}} \pi (ds)dx$, where $\pi (ds)$ is some $\sigma $-finite measure on $S_{d-1}$ and $dx$ is the Lebesgue measure on $\mathbb{R} $. Moreover, the dual process $\widehat{X} $ has the same law as the inversion $(X_{\gamma _t}/\|X_{\gamma _t}\|^2,t\ge 0)$ of $X$, where $\gamma _t$ is the inverse of $t\mapsto \int _0^t\|X\|_s^{-2\alpha }\,ds$. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.
</p>projecteuclid.org/euclid.ejp/1487386998_20170809220241Wed, 09 Aug 2017 22:02 EDTAsymptotics of heights in random trees constructed by aggregationhttp://projecteuclid.org/euclid.ejp/1487646307<strong>Bénédicte Haas</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 25 pp..</p><p><strong>Abstract:</strong><br/>
To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre–existing tree, starting from a segment $T_1$ of length $a_1$. Previous works [5, 10] on that model focus on the influence of $(a_n)$ on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence $(a_n)$ is regularly varying with a non–negative index, so that the sequence $(T_n)$ explodes. We determine the asymptotics of the height of $T_n$ and of the subtrees of $T_n$ spanned by the root and $\ell $ points picked uniformly at random and independently in $T_n$, for all $\ell \in \mathbb N$.
</p>projecteuclid.org/euclid.ejp/1487646307_20170809220241Wed, 09 Aug 2017 22:02 EDTLocal law for the product of independent non-Hermitian random matrices with independent entrieshttp://projecteuclid.org/euclid.ejp/1487991681<strong>Yuriy Nemish</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 35 pp..</p><p><strong>Abstract:</strong><br/>
We consider products of independent square non-Hermitian random matrices. More precisely, let $X_1,\ldots ,X_n$ be independent $N\times N$ random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance $\frac{1} {N}$. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of $n$ random matrices with iid entries converges to \[ \frac{1} {n\pi }1_{|z|\leq 1}|z|^{\frac{2} {n}-2}dz d\overline{z} .\tag{0.1} \] We prove that if the entries of the matrices $X_1,\ldots ,X_n$ are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of $X_1\cdots X_n$ to (0.1) holds up to the scale $N^{-1/2+\varepsilon }$.
</p>projecteuclid.org/euclid.ejp/1487991681_20170809220241Wed, 09 Aug 2017 22:02 EDTFunctional Erdős-Rényi law of large numbers for nonconventional sums under weak dependencehttp://projecteuclid.org/euclid.ejp/1488337348<strong>Yuri Kifer</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 17 pp..</p><p><strong>Abstract:</strong><br/>
We obtain a functional Erdős–Rényi law of large numbers for “nonconventional” sums of the form $\Sigma _n=\sum _{m=1}^n F(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of exponentially fast $\psi $-mixing random vectors and $F$ is a Borel vector function extending in several directions [18] where only i.i.d. random variables $X_1,X_2,...$ were considered.
</p>projecteuclid.org/euclid.ejp/1488337348_20170809220241Wed, 09 Aug 2017 22:02 EDTTransportation–cost inequalities for diffusions driven by Gaussian processeshttp://projecteuclid.org/euclid.ejp/1488596710<strong>Sebastian Riedel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 26 pp..</p><p><strong>Abstract:</strong><br/>
We prove transportation–cost inequalities for the law of SDE solutions driven by general Gaussian processes. Examples include the fractional Brownian motion, but also more general processes like bifractional Brownian motion. In case of multiplicative noise, our main tool is Lyons’ rough paths theory. We also give a new proof of Talagrand’s transportation–cost inequality on Gaussian Fréchet spaces. We finally show that establishing transportation–cost inequalities implies that there is an easy criterion for proving Gaussian tail estimates for functions defined on that space. This result can be seen as a further generalization of the “generalized Fernique theorem” on Gaussian spaces [FH14, Theorem 11.7] used in rough paths theory.
</p>projecteuclid.org/euclid.ejp/1488596710_20170809220241Wed, 09 Aug 2017 22:02 EDTLocal law for random Gram matriceshttp://projecteuclid.org/euclid.ejp/1488942016<strong>Johannes Alt</strong>, <strong>László Erdős</strong>, <strong>Torben Krüger</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 41 pp..</p><p><strong>Abstract:</strong><br/>
We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of $XX^*$.
</p>projecteuclid.org/euclid.ejp/1488942016_20170809220241Wed, 09 Aug 2017 22:02 EDTMeasure-valued Pólya urn processeshttp://projecteuclid.org/euclid.ejp/1490061796<strong>Cécile Mailler</strong>, <strong>Jean-François Marckert</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 33 pp..</p><p><strong>Abstract:</strong><br/>
A Pólya urn process is a Markov chain that models the evolution of an urn containing some coloured balls, the set of possible colours being $\{1,\ldots ,d\}$ for $d\in \mathbb{N} $. At each time step, a random ball is chosen uniformly in the urn. It is replaced in the urn and, if its colour is $c$, $R_{c,j}$ balls of colour $j$ are also added (for all $1\leq j\leq d$).
We introduce a model of measure-valued processes that generalises this construction. This generalisation includes the case when the space of colours is a (possibly infinite) Polish space $\mathcal P$. We see the urn composition at any time step $n$ as a measure ${\cal M}_n$ – possibly non atomic – on $\mathcal P$. In this generalisation, we choose a random colour $c$ according to the probability distribution proportional to ${\cal M}_n$, and add a measure ${\cal R}_c$ in the urn, where the quantity ${\cal R}_c(B)$ of a Borel set $B$ models the added weight of “balls” with colour in $B$.
We study the asymptotic behaviour of these measure-valued Pólya urn processes, and give some conditions on the replacements measures $({\cal R}_c,c\in \mathcal P)$ for the sequence of measures $({\cal M}_n, n\geq 0)$ to converge in distribution, possibly after rescaling. For certain models, related to branching random walks, $({\cal M}_n, n\geq 0)$ is shown to converge almost surely under some moment hypothesis; a particular case of this last result gives the almost sure convergence of the (renormalised) profile of the random recursive tree to a standard Gaussian.
</p>projecteuclid.org/euclid.ejp/1490061796_20170809220241Wed, 09 Aug 2017 22:02 EDTAsymptotics of self-similar growth-fragmentation processeshttp://projecteuclid.org/euclid.ejp/1490061797<strong>Benjamin Dadoun</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 30 pp..</p><p><strong>Abstract:</strong><br/>
Markovian growth-fragmentation processes introduced in [8, 9] extend the pure-fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations [6, 11, 12, 14] when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case [8], we exploit the connection with branching random walks and in particular the martingale convergence of Biggins [18, 19] to derive precise asymptotic estimates. The self-similar case [9] is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed in [10], we obtain limit theorems for empirical measures of the fragments.
</p>projecteuclid.org/euclid.ejp/1490061797_20170809220241Wed, 09 Aug 2017 22:02 EDTOn generalized Gaussian free fields and stochastic homogenizationhttp://projecteuclid.org/euclid.ejp/1490320844<strong>Yu Gu</strong>, <strong>Jean-Christophe Mourrat</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 21 pp..</p><p><strong>Abstract:</strong><br/>
We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an “effective fluctuation tensor” that we denote by $\mathsf{Q} $. We prove an expansion of $\mathsf{Q} $ in the regime of small ellipticity ratio. This expansion shows that the scaling limit of the corrector is not necessarily a classical GFF, and in particular does not necessarily satisfy the Markov property.
</p>projecteuclid.org/euclid.ejp/1490320844_20170809220241Wed, 09 Aug 2017 22:02 EDTBootstrap percolation on products of cycles and complete graphshttp://projecteuclid.org/euclid.ejp/1490320845<strong>Janko Gravner</strong>, <strong>David Sivakoff</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 20 pp..</p><p><strong>Abstract:</strong><br/>
Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta $ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $\theta $ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.
</p>projecteuclid.org/euclid.ejp/1490320845_20170809220241Wed, 09 Aug 2017 22:02 EDTUniversality of random matrices with correlated entrieshttp://projecteuclid.org/euclid.ejp/1490320846<strong>Ziliang Che</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 38 pp..</p><p><strong>Abstract:</strong><br/>
We consider an $N$ by $N$ real symmetric random matrix $X=(x_{ij})$ where $\mathbb{E} [x_{ij}x_{kl}]=\xi _{ijkl}$. Under the assumption that $(\xi _{ijkl})$ is the discretization of a piecewise Lipschitz function and that the correlation is short-ranged we prove that the empirical spectral measure of $X$ converges to a probability measure. The Stieltjes transform of the limiting measure can be obtained by solving a functional equation. Under the slightly stronger assumption that $(x_{ij})$ has a strictly positive definite covariance matrix, we prove a local law for the empirical measure down to the optimal scale $\operatorname{Im} z \gtrsim N^{-1}$. The local law implies delocalization of eigenvectors. As another consequence we prove that the eigenvalue statistics in the bulk agrees with that of the GOE.
</p>projecteuclid.org/euclid.ejp/1490320846_20170809220241Wed, 09 Aug 2017 22:02 EDTReflected Brownian motion: selection, approximation and linearizationhttp://projecteuclid.org/euclid.ejp/1490407496<strong>Marc Arnaudon</strong>, <strong>Xue-Mei Li</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 55 pp..</p><p><strong>Abstract:</strong><br/>
We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process $(W_t)$, the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE’s to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that $(W_t)$ is the weak derivative of a family of reflected Brownian motions with respect to the initial point.
</p>projecteuclid.org/euclid.ejp/1490407496_20170809220241Wed, 09 Aug 2017 22:02 EDTStationary gap distributions for infinite systems of competing Brownian particleshttp://projecteuclid.org/euclid.ejp/1499220068<strong>Andrey Sarantsev</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 20 pp..</p><p><strong>Abstract:</strong><br/>
Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) [PP08] in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.
</p>projecteuclid.org/euclid.ejp/1499220068_20170809220241Wed, 09 Aug 2017 22:02 EDTAveraged vs. quenched large deviations and entropy for random walk in a dynamic random environmenthttp://projecteuclid.org/euclid.ejp/1499306456<strong>Firas Rassoul-Agha</strong>, <strong>Timo Seppäläinen</strong>, <strong>Atilla Yilmaz</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 47 pp..</p><p><strong>Abstract:</strong><br/>
We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears.
</p>projecteuclid.org/euclid.ejp/1499306456_20170809220241Wed, 09 Aug 2017 22:02 EDTExit laws of isotropic diffusions in random environment from large domainshttp://projecteuclid.org/euclid.ejp/1502330523<strong>Benjamin Fehrman</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 37 pp..</p><p><strong>Abstract:</strong><br/>
This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite-range dependence. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [21]. Building upon their work, it is shown by analyzing the associated elliptic boundary-value problem that, almost surely, the smoothed exit law of the diffusion from large domains converges, as the domain’s scale approaches infinity, to that of a Brownian motion. Furthermore, an algebraic rate for the convergence is established in terms of the modulus of the boundary condition.
</p>projecteuclid.org/euclid.ejp/1502330523_20170809220241Wed, 09 Aug 2017 22:02 EDTFinitely dependent insertion processeshttp://projecteuclid.org/euclid.ejp/1491962643<strong>Avi Levy</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 19 pp..</p><p><strong>Abstract:</strong><br/>
A $q$-coloring of $\mathbb Z$ is a random process assigning one of $q$ colors to each integer in such a way that consecutive integers receive distinct colors. A process is $k$-dependent if any two sets of integers separated by a distance greater than $k$ receive independent colorings. Holroyd and Liggett constructed the first stationary $k$-dependent $q$-colorings by introducing an insertion algorithm on the complete graph $K_q$. We extend their construction from complete graphs to weighted directed graphs. We show that complete multipartite analogues of $K_3$ and $K_4$ are the only graphs whose insertion process is finitely dependent and whose insertion algorithm is consistent. In particular, there are no other such graphs among all unweighted graphs and among all loopless complete weighted directed graphs. Similar results hold if the consistency condition is weakened to eventual consistency. Finally we show that the directed de Bruijn graphs of shifts of finite type do not yield $k$-dependent insertion processes, assuming eventual consistency.
</p>projecteuclid.org/euclid.ejp/1491962643_20170810220404Thu, 10 Aug 2017 22:04 EDTAbsorbing-state transition for Stochastic Sandpiles and Activated Random Walkshttp://projecteuclid.org/euclid.ejp/1492070448<strong>Vladas Sidoravicius</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 35 pp..</p><p><strong>Abstract:</strong><br/>
We study the dynamics of two conservative lattice gas models on the infinite $d$-dimensional hypercubic lattice: the Activated Random Walks (ARW) and the Stochastic Sandpiles Model (SSM), introduced in the physics literature in the early nineties. Theoretical arguments and numerical analysis predicted that the ARW and SSM undergo a phase transition between an absorbing phase and an active phase as the initial density crosses a critical threshold. However a rigorous proof of the existence of an absorbing phase was known only for one-dimensional systems. In the present work we establish the existence of such phase transition in any dimension. Moreover, we obtain several quantitative bounds for how fast the activity ceases at a given site or on a finite system. The multi-scale analysis developed here can be extended to other contexts providing an efficient tool to study non-equilibrium phase transitions.
</p>projecteuclid.org/euclid.ejp/1492070448_20170810220404Thu, 10 Aug 2017 22:04 EDTPerturbations of Voter model in one-dimensionhttp://projecteuclid.org/euclid.ejp/1492502428<strong>C.M. Newman</strong>, <strong>K. Ravishankar</strong>, <strong>E. Schertzer</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 42 pp..</p><p><strong>Abstract:</strong><br/>
We study the scaling limit of a large class of voter model perturbations in one dimension, including stochastic Potts models, to a universal limiting object, the continuum voter model perturbation. The perturbations can be described in terms of bulk and boundary nucleations of new colors (opinions). The discrete and continuum (space) models are obtained from their respective duals, the discrete net with killing and Brownian net with killing. These determine the color genealogy by means of reduced graphs . We focus our attention on models where the voter and boundary nucleation dynamics depend only on the colors of nearest neighbor sites, for which convergence of the discrete net with killing to its continuum analog was proved in an earlier paper by the authors. We use some detailed properties of the Brownian net with killing to prove voter model perturbations converge to their continuum counterparts. A crucial property of reduced graphs is that even in the continuum, they are finite almost surely. An important issue is how vertices of the continuum reduced graphs are strongly approximated by their discrete analogues.
</p>projecteuclid.org/euclid.ejp/1492502428_20170810220404Thu, 10 Aug 2017 22:04 EDTGeometry of infinite planar maps with high degreeshttp://projecteuclid.org/euclid.ejp/1492588824<strong>Timothy Budd</strong>, <strong>Nicolas Curien</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 37 pp..</p><p><strong>Abstract:</strong><br/>
We study the geometry of infinite random Boltzmann planar maps with vertices of high degree. These correspond to the duals of the Boltzmann maps associated to a critical weight sequence $(q_{k})_{ k \geq 0}$ for the faces with polynomial decay $k^{-a}$ with $a \in ( \frac{3} {2}, \frac{5} {2})$ which have been studied by Le Gall & Miermont as well as by Borot, Bouttier & Guitter. We show the existence of a phase transition for the geometry of these maps at $a = 2$. In the dilute phase corresponding to $a \in (2, \frac{5} {2})$ we prove that the volume of the ball of radius $r$ (for the graph distance) is of order $r^{ \mathsf{d} }$ with $ \mathsf{d} = (a-\frac 12)/(a-2)$, and we provide distributional scaling limits for the volume and perimeter process. In the dense phase corresponding to $ a \in ( \frac{3} {2},2)$ the volume of the ball of radius $r$ is exponential in $r$. We also study the first-passage percolation (fpp) distance with exponential edge weights and show in particular that in the dense phase the fpp distance between the origin and $\infty $ is finite. The latter implies in addition that the random lattices in the dense phase are transient. The proofs rely on the recent peeling process introduced in [16] and use ideas of [22] in the dilute phase.
</p>projecteuclid.org/euclid.ejp/1492588824_20170810220404Thu, 10 Aug 2017 22:04 EDTA Liouville hyperbolic souvlakihttp://projecteuclid.org/euclid.ejp/1493085635<strong>Johannes Carmesin</strong>, <strong>Bruno Federici</strong>, <strong>Agelos Georgakopoulos</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 19 pp..</p><p><strong>Abstract:</strong><br/>
We construct a transient bounded-degree graph no transient subgraph of which embeds in any surface of finite genus.
Moreover, we construct a transient, Liouville, bounded-degree, Gromov–hyperbolic graph with trivial hyperbolic boundary that has no transient subtree. This answers a question of Benjamini. This graph also yields a (further) counterexample to a conjecture of Benjamini and Schramm. In an appendix by Gábor Pete and Gourab Ray, our construction is extended to yield a unimodular graph with the above properties.
</p>projecteuclid.org/euclid.ejp/1493085635_20170810220404Thu, 10 Aug 2017 22:04 EDTRigorous results for a population model with selection I: evolution of the fitness distributionhttp://projecteuclid.org/euclid.ejp/1493258436<strong>Jason Schweinsberg</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 94 pp..</p><p><strong>Abstract:</strong><br/>
We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each beneficial mutation increases the individual’s fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual’s fitness to give birth. Under certain conditions on the parameters $\mu _N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given time. Our results confirm predictions of Desai and Fisher (2007).
</p>projecteuclid.org/euclid.ejp/1493258436_20170810220404Thu, 10 Aug 2017 22:04 EDTRigorous results for a population model with selection II: genealogy of the populationhttp://projecteuclid.org/euclid.ejp/1493258437<strong>Jason Schweinsberg</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 54 pp..</p><p><strong>Abstract:</strong><br/>
We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each beneficial mutation increases the individual’s fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual’s fitness to give birth. Under certain conditions on the parameters $\mu _N$ and $s_N$, we show that the genealogy of the population can be described by the Bolthausen-Sznitman coalescent. This result confirms predictions of Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).
</p>projecteuclid.org/euclid.ejp/1493258437_20170810220404Thu, 10 Aug 2017 22:04 EDTThe Brownian net and selection in the spatial $\Lambda $-Fleming-Viot processhttp://projecteuclid.org/euclid.ejp/1493345026<strong>Alison Etheridge</strong>, <strong>Nic Freeman</strong>, <strong>Daniel Straulino</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 36 pp..</p><p><strong>Abstract:</strong><br/>
We obtain the Brownian net of [24] as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous gene through such a population.
</p>projecteuclid.org/euclid.ejp/1493345026_20170810220404Thu, 10 Aug 2017 22:04 EDTRicci curvature bounds for weakly interacting Markov chainshttp://projecteuclid.org/euclid.ejp/1493345027<strong>Matthias Erbar</strong>, <strong>Christopher Henderson</strong>, <strong>Georg Menz</strong>, <strong>Prasad Tetali</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 23 pp..</p><p><strong>Abstract:</strong><br/>
We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of spin systems including the Ising and Curie–Weiss models, a class of hard-core models and random walks on groups induced by a conjugacy invariant set of generators.
</p>projecteuclid.org/euclid.ejp/1493345027_20170810220404Thu, 10 Aug 2017 22:04 EDTAn Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension twohttp://projecteuclid.org/euclid.ejp/1493345028<strong>Nils Berglund</strong>, <strong>Giacomo Di Gesù</strong>, <strong>Hendrik Weber</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 27 pp..</p><p><strong>Abstract:</strong><br/>
We study spectral Galerkin approximations of an Allen–Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon } $. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring–Kramers formula for the limiting renormalised stochastic PDE. The effect of the “infinite renormalisation” is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring–Kramers law by a renormalised Carleman–Fredholm determinant.
</p>projecteuclid.org/euclid.ejp/1493345028_20170810220404Thu, 10 Aug 2017 22:04 EDTFluctuations for mean-field interacting age-dependent Hawkes processeshttp://projecteuclid.org/euclid.ejp/1493777018<strong>Julien Chevallier</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 49 pp..</p><p><strong>Abstract:</strong><br/>
The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes $n$ goes to $+\infty $) being granted by the study performed in [9], the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale $n^{-1/2}$) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead of Poisson (which occurs for the law of large numbers limit).
</p>projecteuclid.org/euclid.ejp/1493777018_20170810220404Thu, 10 Aug 2017 22:04 EDTMean-field behavior for nearest-neighbor percolation in $d>10$http://projecteuclid.org/euclid.ejp/1493777019<strong>Robert Fitzner</strong>, <strong>Remco van der Hofstad</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 65 pp..</p><p><strong>Abstract:</strong><br/>
We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as $\gamma =1, \beta =1, \delta =2$. We also prove sharp $x$-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is rigorously proved down from $19$ to $11$. Our results also imply sharp bounds on the critical value of nearest-neighbor percolation on $\mathbb{Z} ^d$, which are provably at most $1.31\%$ off in $d=11$. We make use of the general method analyzed in [17], which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is computer assisted , relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade [25]; and (2) a verification that the numerical conditions in [17] hold true. These two ingredients are implemented in two Mathematica notebooks that can be downloaded from the website of the first author.
The main steps of this paper are (a) to derive a non-backtracking lace expansion for the percolation two-point function; (b) to bound the non-backtracking lace expansion coefficients, thus showing that the general methodology of [17] applies, and (c) to describe the numerical bounds on the coefficients.
</p>projecteuclid.org/euclid.ejp/1493777019_20170810220404Thu, 10 Aug 2017 22:04 EDTExtremes and gaps in sampling from a GEM random discrete distributionhttp://projecteuclid.org/euclid.ejp/1493777020<strong>Jim Pitman</strong>, <strong>Yuri Yakubovich</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 26 pp..</p><p><strong>Abstract:</strong><br/>
We show that in a sample of size $n$ from a $\mathsf{GEM} (0,\theta )$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention $G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an infinite sequence of independent geometric$(i/(i+\theta ))$ variables $G_i$. This extends a known result for the minimum $X_{1:n}$ to other gaps in the range of the sample, and implies that the maximum $X_{n:n}$ has the distribution of $1 + \sum _{i=1}^n G_i$, hence the known result that $X_{n:n}$ grows like $\theta \log (n)$ as $n\to \infty $, with an asymptotically normal distribution. Other consequences include most known formulas for the exact distributions of $\mathsf{GEM} (0,\theta )$ sampling statistics, including the Ewens and Donnelly–Tavaré sampling formulas. For the two-parameter GEM$(\alpha ,\theta )$ distribution we show that the maximal value grows like a random multiple of $n^{\alpha /(1-\alpha )}$ and find the limit distribution of the multiplier.
</p>projecteuclid.org/euclid.ejp/1493777020_20170810220404Thu, 10 Aug 2017 22:04 EDTLong Brownian bridges in hyperbolic spaces converge to Brownian treeshttp://projecteuclid.org/euclid.ejp/1500516020<strong>Xinxin Chen</strong>, <strong>Grégory Miermont</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 15 pp..</p><p><strong>Abstract:</strong><br/>
We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, A property of invariance under re-rooting, The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.
</p>projecteuclid.org/euclid.ejp/1500516020_20170810220404Thu, 10 Aug 2017 22:04 EDTA weak Cramér condition and application to Edgeworth expansionshttp://projecteuclid.org/euclid.ejp/1500516021<strong>Jürgen Angst</strong>, <strong>Guillaume Poly</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 24 pp..</p><p><strong>Abstract:</strong><br/>
We introduce a new, weak Cramér condition on the characteristic function of a random vector which does not only hold for all continuous distributions but also for discrete (non-lattice) ones in a generic sense. We then prove that the normalized sum of independent random vectors satisfying this new condition automatically verifies some small ball estimates and admits a valid Edgeworth expansion for the Kolmogorov metric. The latter results therefore extend the well known theory of Edgeworth expansion under the standard Cramér condition, to distributions that are purely discrete.
</p>projecteuclid.org/euclid.ejp/1500516021_20170810220404Thu, 10 Aug 2017 22:04 EDTEigenvector statistics of sparse random matriceshttp://projecteuclid.org/euclid.ejp/1502417019<strong>Paul Bourgade</strong>, <strong>Jiaoyang Huang</strong>, <strong>Horng-Tzer Yau</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 38 pp..</p><p><strong>Abstract:</strong><br/>
We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erdős-Rényi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged degree increases with the size of the graphs. Our methodology follows [6] by analyzing the eigenvector flow under Dyson Brownian motion, combined with an isotropic local law for Green’s function. As an auxiliary result, we prove that for the eigenvector flow of Dyson Brownian motion with general initial data, the eigenvectors are asymptotically jointly normal in the direction $\boldsymbol q$ after time $\eta _*\ll t\ll r$, if in a window of size $r$, the initial density of states is bounded below and above down to the scale $\eta _*$, and the initial eigenvectors are delocalized in the direction $\boldsymbol q$ down to the scale $\eta _*$.
</p>projecteuclid.org/euclid.ejp/1502417019_20170810220404Thu, 10 Aug 2017 22:04 EDTHitting probabilities of random covering sets in tori and metric spaceshttps://projecteuclid.org/euclid.ejp/1483585523<strong>Esa Järvenpää</strong>, <strong>Maarit Järvenpää</strong>, <strong>Henna Koivusalo</strong>, <strong>Bing Li</strong>, <strong>Ville Suomala</strong>, <strong>Yimin Xiao</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 18 pp..</p><p><strong>Abstract:</strong><br/>
We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in tori, we deal with general analytic generating sets.
</p>projecteuclid.org/euclid.ejp/1483585523_20170821220124Mon, 21 Aug 2017 22:01 EDTQuantitative de Jong theorems in any dimensionhttps://projecteuclid.org/euclid.ejp/1483585524<strong>Christian Döbler</strong>, <strong>Giovanni Peccati</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 35 pp..</p><p><strong>Abstract:</strong><br/>
We develop a new quantitative approach to a multidimensional version of the well-known de Jong’s central limit theorem under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type condition is verified. Our approach allows one to deduce explicit (and presumably optimal) Wasserstein bounds in the case of general $U$-statistics of arbitrary order $d\geq 1$. One of our main findings is that, for vectors of $U$-statistics satisfying de Jong’ s conditions and whose covariances admit a limit, componentwise convergence systematically implies joint convergence to Gaussian: this is the first instance in which such a phenomenon is described outside the frameworks of homogeneous chaoses and of diffusive Markov semigroups.
</p>projecteuclid.org/euclid.ejp/1483585524_20170821220124Mon, 21 Aug 2017 22:01 EDTErratum: Nonlinear filtering for reflecting diffusions in random environments via nonparametric estimationhttps://projecteuclid.org/euclid.ejp/1483585525<strong>Michael A. Kouritzin</strong>, <strong>Wei Sun</strong>, <strong>Jie Xiong</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 2 pp..</p><p><strong>Abstract:</strong><br/>
This is an erratum to EJP paper number 18, volume 9, Nonlinear filtering for reflecting diffusions in random environments via nonparametric estimation.
</p>projecteuclid.org/euclid.ejp/1483585525_20170821220124Mon, 21 Aug 2017 22:01 EDTPathwise uniqueness for an SPDE with Hölder continuous coefficient driven by $\alpha $-stable noisehttps://projecteuclid.org/euclid.ejp/1483585526<strong>Xu Yang</strong>, <strong>Xiaowen Zhou</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 48 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study the pathwise uniqueness of nonnegative solution to the following stochastic partial differential equation with Hölder continuous noise coefficient: \[ \frac{\partial X_t(x)} {\partial t}=\frac{1} {2} \Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L} _t(x),\quad t>0, ~x\in \mathbb{R} , \] where for $1<\alpha <2$ and $0<\beta <1$, $\dot{L} $ denotes an $\alpha $-stable white noise on $\mathbb{R} _+\times \mathbb{R} $ without negative jumps, $G$ satisfies a condition weaker than Lipschitz and $H$ is nondecreasing and $\beta $-Hölder continuous.
For $G\equiv 0$ and $H(x)=x^\beta $, a weak solution to the above stochastic heat equation was constructed in Mytnik (2002) and the pathwise uniqueness of the nonnegative solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of $\alpha $ and $\beta $. In particular, for $\alpha \beta =1$ the solution to the above equation is the density of a super-Brownian motion with $\alpha $-stable branching (see Mytnik (2002)) and our result leads to its pathwise uniqueness for $1<\alpha <\sqrt{5} -1$.
The local Hölder continuity of the solution is also obtained in this paper for fixed time $t>0$.
</p>projecteuclid.org/euclid.ejp/1483585526_20170821220124Mon, 21 Aug 2017 22:01 EDTA central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selectionhttps://projecteuclid.org/euclid.ejp/1484622022<strong>Raphaël Forien</strong>, <strong>Sarah Penington</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 68 pp..</p><p><strong>Abstract:</strong><br/>
We study the evolution of gene frequencies in a population living in $\mathbb{R} ^d$, modelled by the spatial $\Lambda $-Fleming-Viot process with natural selection. We suppose that the population is divided into two genetic types, $a$ and $A$, and consider the proportion of the population which is of type $a$ at each spatial location. If we let both the selection intensity and the fraction of individuals replaced during reproduction events tend to zero, the process can be rescaled so as to converge to the solution to a reaction-diffusion equation (typically the Fisher-KPP equation). We show that the rescaled fluctuations converge in distribution to the solution to a linear stochastic partial differential equation. Depending on whether offspring dispersal is only local or if large scale extinction-recolonization events are allowed to take place, the limiting equation is either the stochastic heat equation with a linear drift term driven by space-time white noise or the corresponding fractional heat equation driven by a coloured noise which is white in time. If individuals are diploid ( i.e. either $AA$, $Aa$ or $aa$) and if natural selection favours heterozygous ($Aa$) individuals, a stable intermediate gene frequency is maintained in the population. We give estimates for the asymptotic effect of random fluctuations around the equilibrium frequency on the local average fitness in the population. In particular, we find that the size of this effect - known as the drift load - depends crucially on the dimension $d$ of the space in which the population evolves, and is reduced relative to the case without spatial structure.
</p>projecteuclid.org/euclid.ejp/1484622022_20170821220124Mon, 21 Aug 2017 22:01 EDTOne-point localization for branching random walk in Pareto environmenthttps://projecteuclid.org/euclid.ejp/1484622023<strong>Marcel Ortgiese</strong>, <strong>Matthew I. Roberts</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 20 pp..</p><p><strong>Abstract:</strong><br/>
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show a very strong form of intermittency, where with high probability most of the mass of the system is concentrated in a single site with high potential. The analogous one-point localization is already known for the parabolic Anderson model, which describes the expected number of particles in the same system. In our case, we rely on very fine estimates for the behaviour of particles near a good point. This complements our earlier results that in the rescaled picture most of the mass is concentrated on a small island.
</p>projecteuclid.org/euclid.ejp/1484622023_20170821220124Mon, 21 Aug 2017 22:01 EDTStochastic differential equations with sticky reflection and boundary diffusionhttps://projecteuclid.org/euclid.ejp/1485486107<strong>Martin Grothaus</strong>, <strong>Robert Voßhall</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 37 pp..</p><p><strong>Abstract:</strong><br/>
We construct diffusion processes in bounded domains $\Omega $ with sticky reflection at the boundary $\Gamma $ in use of Dirichlet forms. In particular, the occupation time on the boundary is positive. The construction covers a static boundary behavior and an optional diffusion along $\Gamma $. The process is a solution to a given SDE for q.e. starting point. Using regularity results for elliptic PDE with Wentzell boundary conditions we show strong Feller properties and characterize the constructed process even for every starting point in $\overline{\Omega } \backslash \Xi $, where $\Xi $ is given explicitly by the involved densities. By a time change we obtain pointwise solutions to SDEs with immediate reflection under weak assumptions on $\Gamma $ and the drift. A non-trivial extension of the construction yields N-particle systems with the stated boundary behavior and singular drifts. Finally, the setting is applied to a model for particles diffusing in a chromatography tube with repulsive interactions.
</p>projecteuclid.org/euclid.ejp/1485486107_20170821220124Mon, 21 Aug 2017 22:01 EDTUniform in time interacting particle approximations for nonlinear equations of Patlak-Keller-Segel typehttps://projecteuclid.org/euclid.ejp/1485831704<strong>Amarjit Budhiraja</strong>, <strong>Wai-Tong Louis Fan</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 37 pp..</p><p><strong>Abstract:</strong><br/>
We study a system of interacting diffusions that models chemotaxis of biological cells or microorganisms (referred to as particles) in a chemical field that is dynamically modified through the collective contributions from the particles. Such systems of reinforced diffusions have been widely studied and their hydrodynamic limits that are nonlinear non-local partial differential equations are usually referred to as Patlak-Keller-Segel (PKS) equations.
Solutions of the classical PKS equation may blow up in finite time and much of the PDE literature has been focused on understanding this blow-up phenomenon. In this work we study a modified form of the PKS equation that is natural for applications and for which global existence and uniqueness of solutions are easily seen to hold. Our focus here is instead on the study of the long time behavior through certain interacting particle systems.
Under the so-called “quasi-stationary hypothesis” on the chemical field, the limit PDE reduces to a parabolic-elliptic system that is closely related to granular media equations whose time asymptotic properties have been extensively studied probabilistically through certain Lyapunov functions [17, 4, 9]. The modified PKS equation studied in the current work is a parabolic-parabolic system for which analogous Lyapunov function constructions are not available. A key challenge in the analysis is that the associated interacting particle system is not a Markov process as the interaction term depends on the whole history of the empirical measure.
We establish, under suitable conditions, uniform in time convergence of the empirical measure of particle states to the solution of the PDE. We also provide uniform in time exponential concentration bounds for rate of the above convergence under additional integrability conditions. Finally, we introduce an Euler discretization scheme for the simulation of the interacting particle system and give error bounds that show that the scheme converges uniformly in time and in the size of the particle system as the discretization parameter approaches zero.
</p>projecteuclid.org/euclid.ejp/1485831704_20170821220124Mon, 21 Aug 2017 22:01 EDTScaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponenthttps://projecteuclid.org/euclid.ejp/1494036159<strong>Ewain Gwynne</strong>, <strong>Xin Sun</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 56 pp..</p><p><strong>Abstract:</strong><br/>
We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various local estimates for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular position in the interior of the first quadrant. We also obtain the exponent for the probability that a word of length $2n$ sampled from the inventory accumulation model corresponds to an empty reduced word, which is equivalent to an asymptotic formula for the partition function of the critical FK planar map model. The estimates of this paper will be used in a subsequent paper to obtain the scaling limit of the lattice walk associated with a finite-volume FK planar map.
</p>projecteuclid.org/euclid.ejp/1494036159_20170821220124Mon, 21 Aug 2017 22:01 EDTWeak error for the Euler scheme approximation of diffusions with non-smooth coefficientshttps://projecteuclid.org/euclid.ejp/1494489631<strong>Valentin Konakov</strong>, <strong>Stéphane Menozzi</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 47 pp..</p><p><strong>Abstract:</strong><br/>
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.
</p>projecteuclid.org/euclid.ejp/1494489631_20170821220124Mon, 21 Aug 2017 22:01 EDTMultifractal analysis for the occupation measure of stable-like processeshttps://projecteuclid.org/euclid.ejp/1496109646<strong>Stéphane Seuret</strong>, <strong>Xiaochuan Yang</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 36 pp..</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the local behavior of the occupation measure $\mu $ of a class of real-valued Markov processes $\mathcal{M} $, defined via a SDE. This (random) measure describes the time spent in each set $A\subset \mathbb{R} $ by the sample paths of $\mathcal{M} $. We compute the multifractal spectrum of $\mu $, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), where the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behavior. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.
</p>projecteuclid.org/euclid.ejp/1496109646_20170821220124Mon, 21 Aug 2017 22:01 EDTRegularity of stochastic kinetic equationshttps://projecteuclid.org/euclid.ejp/1496196076<strong>Ennio Fedrizzi</strong>, <strong>Franco Flandoli</strong>, <strong>Enrico Priola</strong>, <strong>Julien Vovelle</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 42 pp..</p><p><strong>Abstract:</strong><br/>
We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.
</p>projecteuclid.org/euclid.ejp/1496196076_20170821220124Mon, 21 Aug 2017 22:01 EDTPositivity of the time constant in a continuous model of first passage percolationhttps://projecteuclid.org/euclid.ejp/1496196077<strong>Jean-Baptiste Gouéré</strong>, <strong>Marie Théret</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 21 pp..</p><p><strong>Abstract:</strong><br/>
We consider a non trivial Boolean model $\Sigma $ on $\mathbb{R} ^d$ for $d\geq 2$. For every $x,y \in \mathbb{R} ^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $\Sigma $ and at infinite speed inside $\Sigma $. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $\mu \|x\|$ when $\|x\|$ goes to infinity, where $\mu $ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $\mu $. More precisely, under an almost optimal moment assumption on the radii of the balls of the Boolean model, we prove that $\mu >0$ if and only if the intensity $\lambda $ of the Boolean model satisfies $\lambda < \widehat{\lambda } _c$, where $ \widehat{\lambda } _c$ is one of the classical critical parameters defined in continuum percolation.
</p>projecteuclid.org/euclid.ejp/1496196077_20170821220124Mon, 21 Aug 2017 22:01 EDTDisorder relevance without Harris Criterion: the case of pinning model with $\gamma $-stable environmenthttps://projecteuclid.org/euclid.ejp/1497319467<strong>Hubert Lacoin</strong>, <strong>Julien Sohier</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 26 pp..</p><p><strong>Abstract:</strong><br/>
We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha >0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $\alpha >1-\gamma ^{-1}$ there is a shift of the critical point at every temperature whereas when $\alpha < 1-\gamma ^{-1}$, at high temperature the quenched and annealed critical points coincide, and the critical exponents are identical.
</p>projecteuclid.org/euclid.ejp/1497319467_20170821220124Mon, 21 Aug 2017 22:01 EDTErrata to “Processes on unimodular random networks”https://projecteuclid.org/euclid.ejp/1498010464<strong>David Aldous</strong>, <strong>Russell Lyons</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 4 pp..</p><p><strong>Abstract:</strong><br/>
We correct several statements and proofs in our paper, Electron. J. Probab. 12 , Paper 54 (2007), 1454–1508.
</p>projecteuclid.org/euclid.ejp/1498010464_20170821220124Mon, 21 Aug 2017 22:01 EDTLocal circular law for the product of a deterministic matrix with a random matrixhttps://projecteuclid.org/euclid.ejp/1500602612<strong>Haokai Xi</strong>, <strong>Fan Yang</strong>, <strong>Jun Yin</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 77 pp..</p><p><strong>Abstract:</strong><br/>
It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satisfies the circular law. In this paper, we consider the product $TX$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times N$ matrix with independent entries having zero mean and variance $(N\wedge M)^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that $N\sim M$, and the matrix entries $X_{ij}$ have sufficiently high moments. More precisely, if $z$ satisfies $||z|-1|\ge \tau $ for arbitrarily small $\tau >0$, the ESD of $TX$ converges to $\tilde \chi _{\mathbb D}(z) dA(z)$, where $\tilde \chi _{\mathbb D}$ is a rotation-invariant function determined by the singular values of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb C$. The local circular law is valid around $z$ up to scale $(N\wedge M)^{-1/4+\epsilon }$ for any $\epsilon >0$. Moreover, if $|z|>1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N\wedge M)^{-1/2+\epsilon }$ for any $\epsilon >0$.
</p>projecteuclid.org/euclid.ejp/1500602612_20170821220124Mon, 21 Aug 2017 22:01 EDTSpatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noisehttps://projecteuclid.org/euclid.ejp/1503367245<strong>Xia Chen</strong>, <strong>Yaozhong Hu</strong>, <strong>David Nualart</strong>, <strong>Samy Tindel</strong>. <p><strong>Source: </strong>Electronic Journal of Probability, Volume 22, 38 pp..</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter $H \in \left ( \frac 14, \frac 12 \right )$ in the space variable.
</p>projecteuclid.org/euclid.ejp/1503367245_20170821220124Mon, 21 Aug 2017 22:01 EDT