Mostrando recursos 1 - 20 de 168

  1. On the Liouville heat kernel for $k$-coarse MBRW

    Ding, Jian; Zeitouni, Ofer; Zhang, Fuxi
    We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon >0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, \[\exp \left ( - t^{ - \frac 1 { 1 + \frac 1 2 \gamma ^2 } - \varepsilon } \right ) \le p_t^\gamma (x, y) \le \exp \left ( - t^{- \frac 1 { 1...

  2. Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics

    Aru, Juhan; Sepúlveda, Avelio
    We study two-valued local sets, $\mathbb{A} _{-a,b}$, of the two-dimensional continuum Gaussian free field (GFF) with zero boundary condition in simply connected domains. Intuitively, $\mathbb{A} _{-a,b}$ is the (random) set of points connected to the boundary by a path on which the values of the GFF remain in $[-a,b]$. For specific choices of the parameters $a, b$ the two-valued sets have the law of the CLE$_4$ carpet, the law of the union of level lines between all pairs of boundary points, or, conjecturally, the law of the interfaces of the scaling limit of XOR-Ising model. ¶ Two-valued sets are the closure...

  3. The argmin process of random walks, Brownian motion and Lévy processes

    Pitman, Jim; Tang, Wenpin
    In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha _t:=\sup \left \{s \in [0,1]: B_{t+s}=\inf _{u \in [0,1]}B_{t+u} \right \}$ for $t \geq 0$. The argmin process $\alpha $ is stationary, with invariant measure which is arcsine distributed. We prove that $(\alpha _t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot )$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these...

  4. Dynamical freezing in a spin glass system with logarithmic correlations

    Cortines, Aser; Gold, Julian; Louidor, Oren
    We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of...

  5. Sample path properties of permanental processes

    Marcus, Michael B.; Rosen, Jay
    Let $X_{\alpha }=\{X_{\alpha }(t),t\in{\cal T} \}$, $\alpha >0$, be an $\alpha $-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha }$ is a subgaussian process with respect to the metric \[ \sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2} .\nonumber \] This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $\alpha $-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient Lévy processes that are not necessarily symmetric, or...

  6. The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions

    Gheissari, Reza; Lubetzky, Eyal
    We study Swendsen–Wang dynamics for the critical $q$-state Potts model on the square lattice. For $q=2,3,4$, where the phase transition is continuous, the mixing time $t_{\mathrm{mix} }$ is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large $q$, where the phase transition is discontinuous, the authors recently showed that $t_{\mathrm{mix} }$ is highly sensitive to boundary conditions: $t_{\mathrm{mix} } \geq \exp (cn)$ on an $n\times n$ box with periodic boundary, yet under free or monochromatic boundary conditions, $t_{\mathrm{mix} } \leq \exp (n^{o(1)})$. ¶ In this work we classify this effect under boundary conditions...

  7. A support and density theorem for Markovian rough paths

    Chevyrev, Ilya; Ogrodnik, Marcel
    We establish two results concerning a class of geometric rough paths $\mathbf{X} $ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X} $ in $\alpha $-Hölder rough path topology for all $\alpha \in (0,1/2)$, which proves a conjecture of Friz–Victoir [13]. The second is a Hörmander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X} $, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.

  8. Non-equilibrium steady states for networks of oscillators

    Cuneo, Noé; Eckmann, Jean-Pierre; Hairer, Martin; Rey-Bellet, Luc
    Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.

  9. Uniform infinite half-planar quadrangulations with skewness

    Baur, Erich; Richier, Loïc
    We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness ($\mathsf{UIHPQ} _p$ for short, with $p\in [0,1/2]$ measuring the skewness). They interpolate between Kesten’s tree corresponding to $p=0$ and the usual $\mathsf{UIHPQ} $ with a general boundary corresponding to $p=1/2$. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way, and they represent all local limits of (sub)critical Boltzmann quadrangulations whose perimeter tend to infinity. Our main result shows that the...

  10. Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two

    Gold, Julian
    We study the isoperimetric subgraphs of the giant component $\mathbf{{C}} _n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\mathbf{{C}} _n$ with minimal edge boundary to volume ratio. In contrast to the work of [8], the edge boundary is taken only within $\mathbf{{C}} _n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of $\mathbf{{C}} _n$ scales to a deterministic constant, which is itself an isoperimetric...

  11. Resistance growth of branching random networks

    Chen, Dayue; Hu, Yueyun; Lin, Shen
    Consider a rooted infinite Galton–Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $\xi _e$ indexed by all the edges in the tree. We assign the resistance $m^d\,\xi _e$ to each edge $e$ at distance $d$ from the root. In this random electric network, we study the asymptotic behavior of the effective resistance and conductance between the root and the vertices at depth $n$. Our results generalize an existing work of Addario-Berry, Broutin and Lugosi on the binary tree to random branching networks.

  12. Universality of the GOE Tracy-Widom distribution for TASEP with arbitrary particle density

    Ferrari, Patrik L.; Occelli, Alessandra
    We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles $\rho \in (0,1)$. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.

  13. Supermartingale decomposition theorem under $G$-expectation

    Li, Hanwu; Peng, Shige; Song, Yongsheng
    The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales has the decomposition similar to the classical case. The main ideas are to apply the property on uniform continuity of $S_G^\beta (0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.

  14. Coalescent results for diploid exchangeable population models

    Birkner, Matthias; Liu, Huili; Sturm, Anja
    We consider diploid bi-parental analogues of Cannings models: in a population of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from parents $i$ and $j$, where $V=(V_{i,j})_{1\le i \neq j \le N}$ is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an $n$-sample of genes towards a ($\Xi $-)coalescent. This complements Möhle and Sagitov’s (2001)...

  15. Hole probabilities for $\beta $-ensembles and determinantal point processes in the complex plane

    Adhikari, Kartick
    We compute the exact decay rate of the hole probabilities for $\beta $-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory for both processes.

  16. Necessary and sufficient conditions for consistent root reconstruction in Markov models on trees

    Fan, Wai-Tong (Louis); Roch, Sebastien
    We establish necessary and sufficient conditions for consistent root reconstruction in continuous-time Markov models with countable state space on bounded-height trees. Here a root state estimator is said to be consistent if the probability that it returns to the true root state converges to $1$ as the number of leaves tends to infinity. We also derive quantitative bounds on the error of reconstruction. Our results answer a question of Gascuel and Steel [GS10] and have implications for ancestral sequence reconstruction in a classical evolutionary model of nucleotide insertion and deletion [TKF91].

  17. Invariant measures, Hausdorff dimension and dimension drop of some harmonic measures on Galton-Watson trees

    Rousselin, Pierre
    We consider infinite Galton-Watson trees without leaves together with i.i.d. random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in [12], to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\lambda $-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The...

  18. Convergence in distribution norms in the CLT for non identical distributed random variables

    Bally, Vlad; Caramellino, Lucia; Poly, Guillaume
    We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is \[ \varepsilon _{n}(f):={\mathbb{E} }\Big (f\Big (\frac 1{\sqrt n}\sum _{i=1}^{n}Z_{i}\Big )\Big )-{\mathbb{E} }\big (f(G)\big )\rightarrow 0 \] where $Z_{i}$, $i\in \mathbb{N} $, are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity...

  19. 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.

  20. 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...

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