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Register a new userScaling limits in divisible sandpiles: a Fourier multiplier approach
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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.
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The divisible sandpile model is a fixed-energy continuous counterpart of the Abelian sandpile model. We start with a random initial configuration and redistribute mass deterministically. Under certain conditions the sandpile will stabilize. The associated odometer function describes the amount of mass emitted from each vertex during stabilization. In this survey we describe recent scaling limit results of the odometer function depending on different initial configurations and redistribution rules. Moreover we review connections to the obstacle problem from potential theory, including the connection between odometers and limiting shapes of growth models such as iDLA. Finally we state some open problems.
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 establish the existence of free energy limits for several combinatorial models on Erd{o}s-R{e}nyi graph $mathbb {G}(N,lfloor cNrfloor)$ and random $r$-regular graph $mathbb {G}(N,r)$. For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob{a}s and Riordan [Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].
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.
We consider various asymptotic scaling limits $Ntoinfty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with a Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials $Q(zeta)=|zeta|^{2lambda}-(2c/N)log|zeta|$, with $lambda>0$ and $c>-1$, the limiting kernel obeys a linear differential equation of fractional order $1/lambda$ at the origin. For integer $m=1/lambda$ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive the Wards equation for a general class of potentials as a tool to investigate universality. This allows us to determine the functional form of kernels that are translation invariant up to its integration domain.
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