Mostrando recursos 1 - 20 de 1.203

  1. 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...

  2. 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,...

  3. 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.

  4. 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...

  5. 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...

  6. 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...

  7. 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...

  8. 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...

  9. 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...

  10. 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...

  11. 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 $.

  12. 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...

  13. Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation

    Corwin, Ivan; Nica, Mihai
    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...

  14. Intersection and mixing times for reversible chains

    Peres, Yuval; Sauerwald, Thomas; Sousi, Perla; Stauffer, Alexandre
    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 ]}...

  15. Uniqueness of critical Gaussian chaos

    Junnila, Janne; Saksman, Eero
    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.

  16. Double roots of random polynomials with integer coefficients

    Feldheim, Ohad N.; Sen, Arnab
    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...

  17. Mixing time of the fifteen puzzle

    Morris, Ben; Raymer, Anastasia
    We show that the mixing time for the fifteen puzzle in an $n \times n$ torus is on the order of $n^4 \log n$.

  18. Uniform in time interacting particle approximations for nonlinear equations of Patlak-Keller-Segel type

    Budhiraja, Amarjit; Fan, Wai-Tong Louis
    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...

  19. Stochastic differential equations with sticky reflection and boundary diffusion

    Grothaus, Martin; Voßhall, Robert
    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....

  20. One-point localization for branching random walk in Pareto environment

    Ortgiese, Marcel; Roberts, Matthew I.
    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...

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