Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic height distribution in high-dimensional sandpiles
82
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give an asymptotic formula for the single site height distribution of Abelian sandpiles on $mathbb{Z}^d$ as $d to infty$, in terms of $mathsf{Poisson}(1)$ probabilities. We provide error estimates.
rate research
Read More
We study Abelian sandpiles on graphs of the form $G times I$, where $G$ is an arbitrary finite connected graph, and $I subset Z$ is a finite interval. We show that for any fixed $G$ with at least two vertices, the stationary measures $mu_I = mu_{G times I}$ have two extremal weak limit points as $I uparrow Z$. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show that under any of the limiting measures, one can add finitely many grains in such a way that almost surely all sites topple infinitely often. We also show that the extremal limiting measures admit a Markovian coding.
The Hartman-Watson distribution with density $f_r(t)$ is a probability distribution defined on $t geq 0$ which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral $theta(r,t)$ which is difficult to evaluate numerically for small $tto 0$. Using saddle point methods, we obtain the first two terms of the $tto 0$ expansion of $theta(rho/t,t)$ at fixed $rho >0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $rho$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $tto 0$. This has the form $mathbb{P}(frac{1}{t} int_0^t e^{2(B_s+mu s)} ds in da) = (2pi t)^{-1/2} g(a,mu) e^{-frac{1}{t} J(a)} (1 + O(t))$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.
In a recent work Levine et al. (2015) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.
A version of the saddle point method is developed, which allows one to describe exactly the asymptotic behavior of distribution densities of Levy driven stochastic integrals with deterministic kernels. Exact asymptotic behavior is established for (a) the transition probability density of a real-valued Levy process; (b) the transition probability density and the invariant distribution density of a Levy driven Ornstein-Uhlenbeck process; (c) the distribution density of the fractional Levy motion.
In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $(-Delta)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
