Mostrando recursos 1 - 20 de 1.284

  1. Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks

    Sidoravicius, Vladas; Teixeira, Augusto
    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,...

  2. Finitely dependent insertion processes

    Levy, Avi
    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...

  3. Reflected Brownian motion: selection, approximation and linearization

    Arnaudon, Marc; Li, Xue-Mei
    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...

  4. Universality of random matrices with correlated entries

    Che, Ziliang
    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...

  5. Bootstrap percolation on products of cycles and complete graphs

    Gravner, Janko; Sivakoff, David
    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...

  6. On generalized Gaussian free fields and stochastic homogenization

    Gu, Yu; Mourrat, Jean-Christophe
    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...

  7. Asymptotics of self-similar growth-fragmentation processes

    Dadoun, Benjamin
    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...

  8. Measure-valued Pólya urn processes

    Mailler, Cécile; Marckert, Jean-François
    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...

  9. Local law for random Gram matrices

    Alt, Johannes; Erdős, László; Krüger, Torben
    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...

  10. Transportation–cost inequalities for diffusions driven by Gaussian processes

    Riedel, Sebastian
    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,...

  11. Functional Erdős-Rényi law of large numbers for nonconventional sums under weak dependence

    Kifer, Yuri
    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.

  12. Local law for the product of independent non-Hermitian random matrices with independent entries

    Nemish, Yuriy
    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...

  13. Asymptotics of heights in random trees constructed by aggregation

    Haas, Bénédicte
    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...

  14. Inversion, duality and Doob $h$-transforms for self-similar Markov processes

    Alili, Larbi; Chaumont, Loïc; Graczyk, Piotr; Żak, Tomasz
    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...

  15. On asymptotic behavior of the modified Arratia flow

    Konarovskyi, Vitalii
    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...

  16. Local limit of the fixed point forest

    Johnson, Tobias; Schilling, Anne; Slivken, Erik
    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...

  17. Functional central limit theorem for subgraph counting processes

    Owada, Takashi
    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...

  18. Critical window for the configuration model: finite third moment degrees

    Dhara, Souvik; van der Hofstad, Remco; van Leeuwaarden, Johan S.H.; Sen, Sanchayan
    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...

  19. Growth-fragmentation processes and bifurcators

    Shi, Quan
    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 bifurcator, 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 $.

  20. Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

    Kolb, Martin; Savov, Mladen
    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...

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