Mostrando recursos 1 - 20 de 4.133

  1. Erratum to “Scaling for a one-dimensional directed polymer with boundary conditions”

    Seppäläinen, Timo

  2. Limit theorems for point processes under geometric constraints (and topological crackle)

    Owada, Takashi; Adler, Robert J.
    We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. ¶ The main theorems of the paper generate a new class...

  3. Real self-similar processes started from the origin

    Dereich, Steffen; Döring, Leif; Kyprianou, Andreas E.
    Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than $0$ which subsequently was extended to zero initial condition. ¶ For real self-similar Markov processes (rssMps), there is a generalization of Lamperti’s representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. ¶ We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the...

  4. Percolation on the stationary distributions of the voter model

    Ráth, Balázs; Valesin, Daniel
    The voter model on $\mathbb{Z}^{d}$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d\geq3$, the set of (extremal) stationary distributions is a family of measures $\mu_{\alpha}$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_{\alpha}$ is a strongly correlated field of 0’s and 1’s on $\mathbb{Z}^{d}$ in which the density of 1’s is $\alpha$. We consider such a configuration as a site percolation model on $\mathbb{Z}^{d}$. We prove that if $d\geq5$, the probability of existence of an infinite percolation cluster...

  5. Random walks on infinite percolation clusters in models with long-range correlations

    Sapozhnikov, Artem
    For a general class of percolation models with long-range correlations on $\mathbb{Z}^{d}$, $d\geq2$, introduced in [J. Math. Phys. 55 (2014) 083307], we establish regularity conditions of Barlow [Ann. Probab. 32 (2004) 3024–3084] that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincaré inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of [Probab. Theory Related Fields 166 (2016) 619–657] allow to extend...

  6. Fundamental solutions of nonlocal Hörmander’s operators II

    Zhang, Xicheng
    Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$: \[\mathcal{L}^{(\alpha)}_{\sigma,b}f(x):=\mbox{p.v.}\int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\,\mathrm{d}z+b(x)\cdot\nabla f(x)+{\mathscr{L}}f(x),\] where $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are smooth functions and have bounded partial derivatives of all orders greater than $1$, $\delta$ is a small positive number, p.v. stands for the Cauchy principal value and ${\mathscr{L}}$ is a bounded linear operator in Sobolev spaces. Let $B_{1}(x):=\sigma(x)$ and $B_{j+1}(x):=b(x)\cdot\nabla{B}_{j}(x)-\nabla{b(x)}\cdot B_{j}(x)$ for $j\in\mathbb{N}$. Suppose $B_{j}\in C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ for each $j\in\mathbb{N}$. Under the following uniform Hörmander’s type condition: for some $j_{0}\in\mathbb{N}$, \[\inf_{x\in\mathbb{R}^{d}}\inf_{|u|=1}\sum_{j=1}^{j_{0}}|uB_{j}(x)|^{2}>0,\] by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\mathcal{L}^{(\alpha)}_{\sigma,b}$. In particular, we answer a...

  7. KPZ equation limit of higher-spin exclusion processes

    Corwin, Ivan; Tsai, Li-Cheng
    We prove that under a particular weak scaling, the 4-parameter interacting particle system introduced by Corwin and Petrov [Comm. Math. Phys. 343 (2016) 651–700] converges to the Kardar–Parisi–Zhang (KPZ) equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated.

  8. Cutoff for nonbacktracking random walks on sparse random graphs

    Ben-Hamou, Anna; Salez, Justin
    A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous–Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here, we consider nonbacktracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window...

  9. Point-map-probabilities of a point process and Mecke’s invariant measure equation

    Baccelli, François; Haji-Mirsadeghi, Mir-Omid
    A compatible point-shift $F$ maps, in a translation invariant way, each point of a stationary point process $\Phi$ to some point of $\Phi$. It is fully determined by its associated point-map, $f$, which gives the image of the origin by $F$. It was proved by J. Mecke that if $F$ is bijective, then the Palm probability of $\Phi$ is left invariant by the translation of $-f$. The initial question motivating this paper is the following generalization of this invariance result: in the nonbijective case, what probability measures on the set of counting measures are left invariant by the translation of...

  10. Robustness of scale-free spatial networks

    Jacob, Emmanuel; Mörters, Peter
    A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering, we can independently tune the power law exponent $\tau$ of the degree distribution and the rate $-\delta d$ at which the connection probability decreases with the distance of two vertices. We show that the network...

  11. A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions

    Kane, Daniel
    We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_{0}$, which can be decomposed as some function of polynomials $q_{1},\ldots,q_{m}$ with $q_{i}$ normalized and $m=O_{d}(1)$, so that if $X$ is a Gaussian random variable, the probability distribution on $(q_{1}(X),\ldots,q_{m}(X))$ does not have too much mass in any small box. ¶ Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.

  12. Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

    Cuny, Christophe
    We prove that, for (adapted) stationary processes, the so-called Maxwell–Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of $L^{p}$-valued random variables, with $1\le p<\infty$. In this setting, we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [Ann. Probab. 33 (2005) 798–815]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.

  13. Local algorithms for independent sets are half-optimal

    Rahman, Mustazee; Virág, Bálint
    We show that the largest density of factor of i.i.d. independent sets in the $d$-regular tree is asymptotically at most $(\log d)/d$ as $d\to\infty$. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random $d$-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically $2(\log d)/d$. We prove analogous results for Poisson–Galton–Watson trees, which yield bounds for local algorithms on sparse Erdős–Rényi graphs.

  14. A limit theorem for moments in space of the increments of Brownian local time

    Campese, Simon
    We prove a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. [Ann. Prob. 38 (2010) 396–438] and Rosen [Stoch. Dyn. 11 (2011) 5–48], which were later reproven by Hu and Nualart [Electron. Commun. Probab. 15 (2010) 396–410] and Rosen [In Séminaire de Probabilités XLIII (2011) 95–104 Springer] are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space...

  15. Regularity of Wiener functionals under a Hörmander type condition of order one

    Bally, Vlad; Caramellino, Lucia
    We study the local existence and regularity of the density of the law of a functional on the Wiener space which satisfies a criterion that generalizes the Hörmander condition of order one (i.e., involving the first-order Lie brackets) for diffusion processes.

  16. Characterization of cutoff for reversible Markov chains

    Basu, Riddhipratim; Hermon, Jonathan; Peres, Yuval
    A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least $\alpha$, for some $\alpha\in(0,1)$. ¶ We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some “worst” set of stationary measure at least $\alpha$ was not hit by time...

  17. High-dimensional Lipschitz functions are typically flat

    Peled, Ron
    A homomorphism height function on the $d$-dimensional torus $\mathbb{Z}_{n}^{d}$ is a function on the vertices of the torus taking integer values and constrained to have adjacent vertices take adjacent integer values. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. For each of these models, we consider the uniform distribution over all such functions with predetermined values at some fixed vertices (boundary conditions). Our main result is that in high dimensions and with zero boundary values, the random function obtained is typically very flat, having bounded variance at any fixed vertex and taking...

  18. Synchronization by noise for order-preserving random dynamical systems

    Flandoli, Franco; Gess, Benjamin; Scheutzow, Michael
    We provide sufficient conditions for weak synchronization/stabilization by noise for order-preserving random dynamical systems on Polish spaces. That is, under these conditions we prove the existence of a weak point attractor consisting of a single random point. This generalizes previous results in two directions: First, we do not restrict to Banach spaces, and second, we do not require the partial order to be admissible nor normal. As a second main result and application, we prove weak synchronization by noise for stochastic porous media equations with additive noise.

  19. Behavior of the generalized Rosenblatt process at extreme critical exponent values

    Bai, Shuyang; Taqqu, Murad S.
    The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will...

  20. Locality of percolation for Abelian Cayley graphs

    Martineau, Sébastien; Tassion, Vincent
    We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function $\mathrm{p}_{\mathrm{c}}$ defined on the set of Cayley graphs of Abelian groups of rank at least $2$ is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.

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