Mostrando recursos 1 - 20 de 30

  1. Infinite systems of competing Brownian particles

    Sarantsev, Andrey
    Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (Ann....

  2. Spectral measures of factor of i.i.d. processes on vertex-transitive graphs

    Backhausz, Ágnes; Virág, Bálint
    We prove that a measure on $[-d,d]$ is the spectral measure of a factor of i.i.d. process on a vertex-transitive infinite graph if and only if it is absolutely continuous with respect to the spectral measure of the graph. Moreover, we show that the set of spectral measures of factor of i.i.d. processes and that of $\bar{d}_{2}$-limits of factor of i.i.d. processes are the same.

  3. Level lines of the Gaussian free field with general boundary data

    Powell, Ellen; Wu, Hao
    We study the level lines of a Gaussian free field in a planar domain with general boundary data $F$. We show that the level lines exist as continuous curves under the assumption that $F$ is regulated (i.e., admits finite left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the $\operatorname{SLE}_{4}(\underline{\rho})$ process, now with a continuum of force points. A crucial ingredient is a monotonicity property in terms of the boundary data which strengthens a result of Miller and...

  4. Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional

    König, Wolfgang; Mukherjee, Chiranjib
    We study the transformed path measure arising from the self-interaction of a three-dimensional rownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan (Comm. Pure Appl. Math. 505 (1983) 505–528) in terms of a variational formula. Recently (Brownian occupations measures, compactness and large deviations (2014) Preprint) a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers...

  5. Spectral asymptotics for $V$-variable Sierpinski gaskets

    Freiberg, U.; Hambly, B. M.; Hutchinson, John E.
    The family of $V$-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals ($V=1$) and random recursive fractals ($V=\infty$). We consider a class of $V$-variable affine nested fractals based on the Sierpinski gasket with a general class of measures. We calculate the spectral exponent for a general measure and find the spectral dimension for these fractals. We show that the spectral properties and on-diagonal heat kernel estimates for $V$-variable fractals are closer to those of scale irregular fractals, in that it is the fluctuations in scale that determine their...

  6. Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?

    Ahlberg, Daniel; Steif, Jeffrey E.
    Consider a monotone Boolean function $f:\{0,1\}^{n}\to\{0,1\}$ and the canonical monotone coupling $\{\eta_{p}:p\in[0,1]\}$ of an element in $\{0,1\}^{n}$ chosen according to product measure with intensity $p\in[0,1]$. The random point $p\in[0,1]$ where $f(\eta_{p})$ flips from $0$ to $1$ is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large $n$, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions...

  7. Edgeworth expansions for profiles of lattice branching random walks

    Grübel, Rudolf; Kabluchko, Zakhar
    Consider a branching random walk on $\mathbb{Z}$ in discrete time. Denote by $L_{n}(k)$ the number of particles at site $k\in\mathbb{Z}$ at time $n\in\mathbb{N}_{0}$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_{n}(k)$. We establish the following asymptotic expansion of $L_{n}(k)$, as $n\to\infty$: \begin{equation*}\mathrm{e}^{-\varphi(0)n}L_{n}(k)=\frac{\mathrm{e}^{-\frac{1}{2}x_{n}^{2}(k)}}{\sqrt{2\pi\varphi"(0)n}}\sum_{j=0}^{r}\frac{F_{j}(x_{n}(k))}{n^{j/2}}+o(n^{-\frac{r+1}{2}})\quad \text{a.s.},\end{equation*} where $r\in\mathbb{N}_{0}$ is arbitrary, $\varphi(\beta)=\log\sum_{k\in\mathbb{Z}}\mathrm{e}^{\beta k}\mathbb{E}L_{1}(k)$ is the cumulant generating function of the intensity of the branching random walk and \begin{equation*}x_{n}(k)=\frac{k-\varphi'(0)n}{\sqrt{\varphi"(0)n}}.\end{equation*} The expansion is valid uniformly in $k\in\mathbb{Z}$ with probability $1$ and the $F_{j}$’s are polynomials whose random coefficients can be expressed through the derivatives of $\varphi$ and...

  8. Pólya tree posterior distributions on densities

    Castillo, Ismaël
    Pólya trees form a popular class of prior distributions used in Bayesian nonparametrics. For some choice of parameters, Pólya trees are prior distributions on density functions. In this paper we carry out a frequentist analysis of the induced posterior distributions in the density estimation model. We investigate the contraction rate of Pólya tree posterior densities in terms of the supremum loss and study the limiting shape distribution. A nonparametric Bernstein–von Mises theorem is established, as well as a Bayesian Donsker theorem for the posterior cumulative distribution function.

  9. Doubly probabilistic representation for the stochastic porous media type equation

    Barbu, Viorel; Röckner, Michael; Russo, Francesco
    The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space $\mathbb{R}^{1}$ perturbed by a multiplicative colored noise. For almost all random realizations $\omega$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion.

  10. The power of averaging at two consecutive time steps: Proof of a mixing conjecture by Aldous and Fill

    Hermon, Jonathan; Peres, Yuval
    Let $(X_{t})_{t=0}^{\infty}$ be an irreducible reversible discrete-time Markov chain on a finite state space $\Omega$. Denote its transition matrix by $P$. To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain $(X_{t}^{\mathrm{c}})_{t\ge0}$ whose kernel is given by $H_{t}:=e^{-t}\sum_{k}(tP)^{k}/k!$. Another possibility is to consider the associated averaged chain $(X_{t}^{\mathrm{ave}})_{t=0}^{\infty}$, whose distribution at time $t$ is obtained by replacing $P^{t}$ by $A_{t}:=(P^{t}+P^{t+1})/2$. ¶ A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let $(X_{t}^{(n)})_{t=0}^{\infty}$ be a sequence of irreducible reversible discrete-time...

  11. Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble

    Nguyen, Gia Bao; Remenik, Daniel
    We show that the squared maximal height of the top path among $N$ non-intersecting Brownian bridges starting and ending at the origin is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a pre-asymptotic version of K. Johansson’s result (Comm. Math. Phys. 242 (2003) 277–329) that the supremum of the $\operatorname{Airy}_{2}$ process minus a parabola has the Tracy–Widom GOE distribution, and as such it provides an explanation for how this distribution arises in models belonging to the KPZ universality class with flat initial data. The result can...

  12. A boundedness trichotomy for the stochastic heat equation

    Chen, Le; Khoshnevisan, Davar; Kim, Kunwoo
    We consider the stochastic heat equation with a multiplicative white noise forcing term under standard “intermitency conditions.” The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution $x\mapsto u(t,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_{0}$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\mathbf{\Lambda}:=\lim_{\vert x\vert \to\infty}\vert \log u_{0}(x)\vert /(\log\vert x\vert )^{2/3}$.

  13. Integration formulas for Brownian motion on classical compact Lie groups

    Dahlqvist, Antoine
    Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur–Weyl dualities based on stochastic calculus.

  14. A discrete log-Sobolev inequality under a Bakry–Émery type condition

    Johnson, Oliver
    We consider probability mass functions $V$ supported on the positive integers using arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry–Émery condition for a Markov birth and death operator with invariant measure $V$. Under this condition, we prove a new modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.

  15. Limit theorems for longest monotone subsequences in random Mallows permutations

    Basu, Riddhipratim; Bhatnagar, Nayantara
    We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $\pi$ is proportional to $q^{\operatorname{inv}(\pi)}$ where $q$ is a positive parameter and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$. ¶ In our main result we show that when $0

  16. Crossing probabilities for critical Bernoulli percolation on slabs

    Basu, Deepan; Sapozhnikov, Artem
    We prove that in the critical Bernoulli percolation on graphs $\mathbb{Z}^{2}\times\{0,\ldots,k-1\}^{d-2}$, for each $\rho>0$, the probability of open left-right crossing of rectangle $[0,\rho N]\times[0,N]\times[0,k-1]^{d-2}$ is uniformly positive.

  17. Joint convergence of random quadrangulations and their cores

    Addario-Berry, Louigi; Wen, Yuting
    We show that a uniform quadrangulation, its largest $2$-connected block, and its largest simple block jointly converge to the same Brownian map in distribution for the Gromov–Hausdorff–Prokhorov topology. We start by deriving a local limit theorem for the asymptotics of maximal block sizes, extending the result in (Random Structures Algorithms 19 (2001) 194–246). The resulting diameter bounds for pendant submaps of random quadrangulations straightforwardly lead to Gromov–Hausdorff convergence. To extend the convergence to the Gromov–Hausdorff–Prokhorov topology, we show that exchangeable “uniformly asymptotically negligible” attachments of mass simply yield, in the limit, a deterministic scaling of the mass measure.

  18. Brownian motion correlation in the peanosphere for $\kappa>8$

    Gwynne, Ewain; Holden, Nina; Miller, Jason; Sun, Xin
    The peanosphere (or “mating of trees”) construction of Duplantier, Miller, and Sheffield encodes certain types of $\gamma$-Liouville quantum gravity (LQG) surfaces ($\gamma\in(0,2)$) decorated with an independent $\operatorname{SLE}_{\kappa}$ ($\kappa=16/\gamma^{2}>4$) in terms of a correlated two-dimensional Brownian motion and provides a framework for showing that random planar maps decorated with statistical physics models converge to LQG decorated with an $\operatorname{SLE}$. Previously, the correlation for the Brownian motion was only explicitly identified as $-\cos(4\pi/\kappa)$ for $\kappa\in(4,8]$ and unknown for $\kappa>8$. The main result of this work is that this formula holds for all $\kappa>4$. This supplies the missing ingredient for proving convergence results...

  19. Limit theorems for the left random walk on $\operatorname{GL}_{d}(\mathbb{R})$

    Cuny, Christophe; Dedecker, Jérôme; Jan, Christophe
    Motivated by a recent work of Benoist and Quint and extending results from the PhD thesis of the third author, we obtain limit theorems for products of independent and identically distributed elements of $\operatorname{GL}_{d}(\mathbb{R})$, such as the Marcinkiewicz–Zygmund strong law of large numbers, the CLT (with rates in Wasserstein’s distances) and almost sure invariance principles with rates.

  20. A rice method proof of the null-space property over the Grassmannian

    Azaïs, Jean-Marc; Mourareau, Stéphane; De Castro, Yohann
    The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. ¶ Although this property is the keystone of $\ell_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact,...

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