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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.
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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.
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.
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.
We show that in abelian sandpiles on infinite Galton-Watson trees, the probability that the total avalanche has more than $t$ topplings decays as $t^{-1/2}$. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (2003), that was previously used by Lyons, Morris and Schramm (2008) to study uniform spanning forests on $mathbb{Z}^d$, $dgeq 3$, and other transient graphs.
We explore squeezed coherent states of a 3-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered $X_1$-Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three mode squeezed coherent states resolve the identity operator. We evaluate Mandels $Q$-parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Further, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate Wigner function of these three mode squeezed coherent states and demonstrate squeezing property explicitly.
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