Mostrando recursos 1 - 20 de 150

  1. Quantitative estimates for the flux of TASEP with dilute site disorder

    Bahadoran, C.; Bodineau, T.
    We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. For high dilution, we obtain an accurate description of the flux. The result is established under a decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum. To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.

  2. Excited random walk in a Markovian environment

    Travers, Nicholas F.
    One dimensional excited random walk has been extensively studied for bounded, i.i.d. cookie environments. In this case, many important properties of the walk including transience or recurrence, positivity or non-positivity of the speed, and the limiting distribution of the position of the walker are all characterized by a single parameter $\delta $, the total expected drift per site. In the more general case of stationary ergodic environments, things are not so well understood. If all cookies are positive then the same threshold for transience vs. recurrence holds, even if the cookie stacks are unbounded. However, it is unknown if the...

  3. Pathwise construction of tree-valued Fleming-Viot processes

    Gufler, Stephan
    In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued $\Xi $-Fleming-Viot processes and deduce path properties. For instance, these processes usually have a. s. càdlàg paths with jumps at the times of large reproduction events. In the case of coming down from infinity, the construction...

  4. A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes

    Gufler, Stephan
    We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of tree-valued Fleming-Viot processes from a $\Xi $-lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.

  5. Extremes of local times for simple random walks on symmetric trees

    Abe, Yoshihiro
    We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim _{n \to \infty } r_n = \infty $ and $\limsup _{n \to \infty } r_n/n <1$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty } (dx) \otimes e^{-2\sqrt{\log b} ~y}dy$, where $\alpha > 0$...

  6. Vertex reinforced non-backtracking random walks: an example of path formation

    Le Goff, Line C.; Raimond, Olivier
    This article studies vertex reinforced random walks that are non-backtracking (denoted VRNBW), i.e. U-turns forbidden. With this last property and for a strong reinforcement, the emergence of a path may occur with positive probability. These walks are thus useful to model the path formation phenomenon, observed for example in ant colonies. This study is carried out in two steps. First, a large class of reinforced random walks is introduced and results on the asymptotic behavior of these processes are proved. Second, these results are applied to VRNBWs on complete graphs and for reinforced weights $W(k)=k^\alpha $, with $\alpha \ge 1$....

  7. Characterizing stationary 1+1 dimensional lattice polymer models

    Chaumont, Hans; Noack, Christian
    Motivated by the study of directed polymer models with random weights on the square integer lattice, we define an integrability property shared by the log-gamma, strict-weak, beta, and inverse-beta models. This integrability property encapsulates a preservation in distribution of ratios of partition functions which in turn implies the so called Burke property. We show that under some regularity assumptions, up to trivial modifications, there exist no other models possessing this property.

  8. Sublinearity of the number of semi-infinite branches for geometric random trees

    Coupier, David
    The present paper addresses the following question: for a geometric random tree in $\mathbb{R} ^{2}$, how many semi-infinite branches cross the circle $\mathcal{C} _{r}$ centered at the origin and with a large radius $r$? We develop a method ensuring that the expectation of the number $\chi _{r}$ of these semi-infinite branches is $o(r)$. The result follows from the fact that, far from the origin, the distribution of the tree is close to that of an appropriate directed forest which lacks bi-infinite paths. In order to illustrate its robustness, the method is applied to three different models: the Radial Poisson Tree...

  9. Fourth moment theorems on the Poisson space in any dimension

    Döbler, Christian; Vidotto, Anna; Zheng, Guangqu
    We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. Döbler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case. ¶ Finally, a transfer principle “from-Poisson-to-Gaussian” is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.

  10. Decomposition of mean-field Gibbs distributions into product measures

    Eldan, Ronen; Gross, Renan
    We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are critical points of an associated mean-field functional. This extends a previous work by the first author. As an application, we demonstrate how this framework helps characterize both Ising models satisfying a mean-field condition and the conditional distributions which arise in the emerging theory of nonlinear large deviations, both in the dense case and in the polynomially-sparse case.

  11. Moment convergence of balanced Pólya processes

    Janson, Svante; Pouyanne, Nicolas
    It is known that in an irreducible small Pólya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also is balanced, this normal convergence holds with convergence of all moments, thus giving asymptotics of (central) moments.

  12. Circular law for the sum of random permutation matrices

    Basak, Anirban; Cook, Nicholas; Zeitouni, Ofer
    Let $P_n^1,\dots , P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum _{\ell =1}^d P_n^\ell $. We show that if $\log ^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d} $ converges weakly to the circular law in probability as $n \to \infty $.

  13. Exponential concentration of cover times

    Zhai, Alex
    We prove an exponential concentration bound for cover times of general graphs in terms of the Gaussian free field, extending the work of Ding, Lee, and Peres [8] and Ding [7]. The estimate is asymptotically sharp as the ratio of hitting time to cover time goes to zero. ¶ The bounds are obtained by showing a stochastic domination in the generalized second Ray-Knight theorem, which was shown to imply exponential concentration of cover times by Ding in [7]. This stochastic domination result appeared earlier in a preprint of Lupu [22], but the connection to cover times was not mentioned.

  14. Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation

    Powell, Ellen
    We show that, for general convolution approximations to a large class of log-correlated fields, including the 2d Gaussian free field, the critical chaos measures with derivative normalisation converge to a limiting measure $\mu '$. This limiting measure does not depend on the choice of approximation. Moreover, it is equal to the measure obtained using the Seneta–Heyde renormalisation at criticality, or using a white-noise approximation to the field.

  15. Localization of directed polymers with general reference walk

    Bates, Erik
    Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an $\alpha $-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer’s endpoint distribution is asymptotically purely atomic,...

  16. Affine processes with compact state space

    Krühner, Paul; Larsson, Martin
    The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove in particular that (i) no diffusion is possible; (ii) jumps are possible and enforce a grid-like structure of the state space; (iii) jump components can feed into drift components, but not vice versa. Using our main structural theorem, we classify all bivariate affine processes with compact state space. Unlike the classical case, the characteristic function of an affine process with compact state space may vanish, even...

  17. The Schrödinger equation with spatial white noise potential

    Debussche, Arnaud; Weber, Hendrik
    We consider the linear and nonlinear Schrödinger equation with a spatial white noise as a potential over the two dimensional torus. We prove existence and uniqueness of solutions to an initial value problem for suitable initial data. Our construction is based on a change of unknown originally used in [13] and conserved quantities.

  18. Recurrence and transience of contractive autoregressive processes and related Markov chains

    Zerner, Martin P.W.
    We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.

  19. A random walk on the symmetric group generated by random involutions

    Bernstein, Megan
    The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this paper to mix for $\frac{1} {2} \leq p \leq 1$ fixed, $n$ sufficiently large in between $\log _{1/p}(n)$ steps and $\log _{2/(1+p)}(n)$ steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. Monotonicity relations used in the...

  20. On sensitivity of mixing times and cutoff

    Hermon, Jonathan; Peres, Yuval
    A sequence of chains exhibits (total variation) cutoff (resp., pre-cutoff) if for all $0<\epsilon < 1/2$, the ratio $t_{\mathrm{mix} }^{(n)}(\epsilon )/t_{\mathrm{mix} }^{(n)}(1-\epsilon )$ tends to 1 as $n \to \infty $ (resp., the $\limsup $ of this ratio is bounded uniformly in $\epsilon $), where $t_{\mathrm{mix} }^{(n)}(\epsilon )$ is the $\epsilon $-total variation mixing time of the $n$th chain in the sequence. We construct a sequence of bounded degree graphs $G_n$, such that the lazy simple random walks (LSRW) on $G_n$ satisfy the “product condition” $\mathrm{gap} (G_n) t_{\mathrm{mix} }^{(n)}(\epsilon ) \to \infty $ as $n \to \infty $, where $\mathrm{gap}...

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