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Vol 214, No 12 (2023)
Convergence of a sandpile model on a triangular lattice
Abstract
We present a survey of results on convergence in sandpile models. For a sandpile model on a triangular lattice we prove results similar to the ones known for a square lattice. Namely, consider the sandpile model on the integer points of the plane and put n grains of sand at the origin. Let us begin the process of relaxation: if the number of grains of sand at some vertex z is not less than its valency (in this case we say that the vertex z is unstable), then we move a grain of sand from z to each adjacent vertex, and then repeat this operation as long as there are unstable vertices. We prove that the support of the state (nδ0)∘ in which the process stabilizes grows at a rate of √n and, after rescaling with coefficient √n, (nδ0)∘ has a limit in the weak-∗ topology.
This result was established by Pegden and Smart for the square lattice (where every vertex is connected with four nearest neighbours); we extend it to a triangular lattice (where every vertex is connected with six neighbours).
3-25
Estimates for integrals of derivatives of $n$-valent functions and geometric properties of domains
Abstract
A number of questions concerning the behaviour of double integrals of the moduli of the derivatives of bounded n-valent functions and, in particular, of rational functions of fixed degree n are considered. For domains with rectifiable boundaries the sharp order of growth of such integral means is found in its dependence on n. Upper bounds for domains with fractal boundaries are obtained, which depend on the Minkowski dimension of the boundary of the domain. In certain cases these bounds are shown to be close to sharp ones. Lower bounds in terms of the integral means spectra of conformal mappings are also found. These inequalities refine Dolzhenko's classical results (1966) and some recent results due to the authors.
26-45
The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\mathbb C^{\ast})^n$ of the Grassmann manifolds $G_{n,2}$
Abstract
The complex Grassmann manifolds Gn,k appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case k=2 is of special interest on its own as the manifolds Gn,2 have several remarkable properties which distinguish them from the Gn,k for k>2.
In our paper we obtain results which, essentially using the specifics of the Grassmann manifolds Gn,2, develop connections between algebraic geometry and equivariant topology. They are related to well-known problems of the canonical action of the algebraic torus (C∗)n on Gn,2 and the induced action of the compact torus Tn⊂(C∗)n.
Kapranov proved that the Deligne-Mumford-Grothendieck-Knudsen compactification ¯M(0,n) of the space of n-pointed rational stable curves can be realized as the Chow quotient Gn,2//(C∗)n. In recent papers of the authors a constructive description of the orbit space Gn,2/Tn was obtained. In deducing this result the notions of the complex of admissible polytopes and the universal space of parameters Fn for the Tn-action on Gn,2 were of essential use.
Using the techniques of wonderful compactification, in this paper an explicit construction of the space Fn is presented. In combination with Keel's description of ¯M(0,n), this construction enabled one to obtain an explicit diffeomorphism between Fn and ¯M(0,n). In this way, we give a description of Gn,2//(C∗)n as the space Fn with a structure described in terms of admissible polytopes Pσ and spaces Fσ.
46-75
On the spectrum of Landau Hamiltonian perturbed by a periodic electric potential
Abstract
We study the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential V∈L2loc(R2;R) assuming that the magnetic flux of the homogeneous magnetic field B>0 satisfies the condition (2π)−1Bv(K)=Q−1, Q∈N, where v(K) is the area of the unit cell K of the period lattice of the potential V. For arbitrary periodic potentials V∈L2loc(R2;R) with zero mean V0=0 we show that the spectrum has no eigenvalues different from Landau levels. For periodic potentials V∈L2loc(R2;R)∖C∞(R2;R) we also show that the spectrum is absolutely continuous.
76-105
Infinite elliptic hypergeometric series: convergence and diffrence equations
Abstract
We derive finite difference equations of infinite order for theta-hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, and we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion of the convergence of q-hypergeometric series for |q|=1, qn≠1, to the elliptic level and prove the convergence of infinite very-well poised elliptic hypergeometric r+1Vr-series for restricted values of q.
106-134
Dual exceptional collections on Lagrangian Grassmannians
Abstract
We construct graded left dual exceptional collections to the exceptional collections generating the Kuznetsov-Polishchuk blocks on Lagrangian Grassmannians. As an application, we find explicit resolutions for some natural irreducible equivariant vector bundles.
135-158