Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

Chatterjee, Sourav; Kirkpatrick, Kay

doi:10.1002/cpa.21388

Cited by 19 publications

(41 citation statements)

References 37 publications

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“…In addition to spin glass theory and random matrix theory, the idea of using COM to derive low-dimensional nonlinear equations to replace linear equations in high dimensions is helpful in a variety of contexts. 6 The mixed matrix moments for the complex Ginibre ensemble are particularly nice moments to consider because their combinatorics is as simple as possible. (Indeed it is somewhat simpler than the usual Catalan numbers that arise in the GUE/GOE (Gaussian unitary/orthogonal ensemble) moments or the bipartite Catalan numbers that arise in the Marčenko-Pastur law.)…”

Section: Discussionmentioning

confidence: 99%

A note on mixed matrix moments for the complex Ginibre ensemble

Walters

Starr

2015

Journal of Mathematical Physics

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Articles you may be interested inEmbedded Gaussian unitary ensembles with U(Ω)⊗SU(r) embedding generated by random twobody interactions with SU(r) symmetryWe consider the mixed matrix moments for the complex Ginibre ensemble. These are well-known. We consider the relation to the expected overlap functions of Chalker and Mehlig. This leads to new asymptotic problems for the overlap. We obtain some results, but we also state some remaining open problems. C 2015 AIP Publishing LLC. FIG. 1. Some analogous elements in random matrix and spin glass theory. (Proofs may differ considerably.) 013301-3 M. Walters and S. Starr J. Math. Phys. 56, 013301 (2015)Another easy fact is that, due to symmetry, m n (p, q) = 0 unless p(1) + · · · + p(k) = q(1) + · · · + q(k). And, of course, m 0 = 1. Using this and the method of induction it is easy to prove Theorem 2.1.

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Section: Discussionmentioning

confidence: 99%

A note on mixed matrix moments for the complex Ginibre ensemble

Walters

Starr

2015

Journal of Mathematical Physics

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“…On one side, Nelson's derivation is based on a variational principle, which makes his approach more attractive. However, although different formulations of the NLS on graphs have been introduced in physics and mathematics [8,11,15,16,29], not much is known through Nelson's approach, mainly because the theory of discrete optimal transport has not been seriously explored until the past few years [12,24,26]. On the other side, most of the discrete formulations for the NLS, especially those defined on lattices, are obtained by discretizations of the continuous NLS.…”

Section: Introductionmentioning

confidence: 99%

A discrete Schrödinger equation via optimal transport on graphs

Chow

Zhou

2019

Journal of Functional Analysis

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In 1966, Edward Nelson presented an interesting derivation of the Schrödinger equation using Brownian motion. Recently, this derivation is linked to the theory of optimal transport, which shows that the Schrödinger equation is a Hamiltonian system on the probability density manifold equipped with the Wasserstein metric. In this paper, we consider similar matters on a finite graph. By using discrete optimal transport and its corresponding Nelson's approach, we derive a discrete Schrödinger equation on a finite graph. The proposed system is quite different from the commonly referred discretized Schrödinger equations. It is a system of nonlinear ordinary differential equations (ODEs) with many desirable properties. Several numerical examples are presented to illustrate the properties. R d W(x, y)ρ(t, y)dy = 0 ,

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“…The proof of the equivalence of ensembles in [14] is nice and rather straightforward, but is based on a local central limit theorem as is actually the proof in [6]. We do not know whether such a result is valid for our set-up (the parity constraint in the definition (4) below indicates that the statement should be rather specific). Instead, we exploit the huge degeneracy of the spectrum of H pot to construct explicitly a set of states in the micro-canonical ensemble with an entropy that approaches the canonical entropy of the Gibbs state in the thermodynamic limit, see Lemma 2 below.…”

Section: Introductionmentioning

confidence: 99%

Equivalence of Ensembles, Condensation and Glassy Dynamics in the Bose–Hubbard Hamiltonian

Huveneers¹,

Theil²

2019

J Stat Phys

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We study mathematically the equilibrium properties of the Bose-Hubbard Hamiltonian in the limit of a vanishing hopping amplitude. This system conserves the energy and the number of particles. We establish the equivalence between the microcanonical and the grand-canonical ensembles for all allowed values of the density of particles ρ and density of energy ε. Moreover, given ρ, we show that the system undergoes a transition as ε increases, from a usual positive temperature state to the infinite temperature state where a macroscopic excess of energy condensates on a single site. Analogous results have been obtained by S. Chatterjee [6] for a closely related model. We introduce here a different method to tackle this problem, hoping that it reflects more directly the basic understanding stemming from statistical mechanics. We discuss also how, and in which sense, the condensation of energy leads to a glassy dynamics. arXiv:1903.12438v1 [cond-mat.stat-mech]

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Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

Abstract: We show that the thermodynamics of the focusing cubic discrete nonlinear Schrö-dinger equation are exactly solvable in dimension 3 and higher. A number of explicit formulas are derived.

Cited by 19 publications

References 37 publications

A note on mixed matrix moments for the complex Ginibre ensemble

A note on mixed matrix moments for the complex Ginibre ensemble

A discrete Schrödinger equation via optimal transport on graphs

Equivalence of Ensembles, Condensation and Glassy Dynamics in the Bose–Hubbard Hamiltonian

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