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Register a new userConstructing fractional Gaussian fields from long-range divisible sandpiles on the torus
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In cite{Cipriani2016}, the authors proved that, with the appropriate rescaling, the odometer of the (nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we study $alpha$-long-range divisible sandpiles, similar to those introduced in cite{Frometa2018}. We show that, for $alpha in (0,2)$, the limiting field is a fractional Gaussian field on the torus with parameter $alpha/2$. However, for $alpha in [2,infty)$, we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on the generator of Levy walks. The central tool for obtaining our results is a careful study of the spectrum of the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence of the eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtain precise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine the order of the expected maximum of the discrete fractional Gaussian field with parameter $gamma=min {alpha,2}$ and $alpha in mathbb{R}_+backslash{2}$ on a finite grid.
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Inspired by Fr{o}hlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $mathbb{Z}^d$, $dgeq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $alpha>d$ and improves previous results obtained by Park for $alpha>3d+1$, and by Ginibre, Grossmann, and Ruelle for $alpha> d+1$, where $alpha$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomial decaying magnetic field with power $delta>0$ as $h^*|x|^{-delta}$, where $h^* >0$. For $d<alpha<d+1$, the phase transition occurs when $delta>d-alpha$, and when $h^*$ is small enough over the critical line $delta=d-alpha$. For $alpha geq d+1$, $delta>1$ it is enough to prove the phase transition, and for $delta=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.
We consider a Stark Hamiltonian on a two-dimensional bounded domain with Dirichlet boundary conditions. In the strong electric field limit we derive, under certain local convexity conditions, a three-term asymptotic expansion of the low-lying eigenvalues. This shows that the excitation frequencies are proportional to the square root of the boundary curvature at a certain point determined by the direction of the electric field.
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.
In this paper we continue the formal analysis of the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation that we began in [18]. They have the form $fleft( x,v,tright) =gleft(v-Lleft( tright) x,tright) $ where $Lleft( tright) =Aleft(I+tAright) ^{-1}$ where $A$ is a constant matrix. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. Depending on the properties of the collision kernel the collision and the hyperbolic terms might be of the same order of magnitude as $ttoinfty$, or the collision term could be the dominant one for large times, or the hyperbolic term could be the largest. The first case has been rigorously studied in [17]. Formal asymptotic expansions in the second case have been obtained in [18]. All the solutions obtained in this case can be approximated by Maxwellian distributions with changing temperature. In this paper we focus in the case where the hyperbolic terms are much larger than the collision term for large times (hyperbolic-dominated behavior). In the hyperbolic-dominated case it does not seem to be possible to describe in a simple way all the long time asymptotics of the solutions, but we discuss several physical situations and formulate precise conjectures. We give explicit formulas for the relationship between density, temperature and entropy for these solutions. These formulas differ greatly from the ones at equilibrium.
In this paper, we prove a power-law version dynamical localization for a random operator $mathrm{H}_{omega}$ on $mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schrodinger equation $$mathrm{i}partial_{t}u=mathrm{H}_{omega}u, quad u in ell^2(mathbb{Z}^d), $$ the Sobolev norm of the solution with well localized initial state is bounded for any $tgeq 0$.
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