[Paper Review] A Matrix model for plane partitions and TASEP

Bertrand Eynard|arXiv (Cornell University)|May 5, 2009

Stochastic processes and statistical mechanics11 citations

TL;DR

This paper introduces an exact matrix model formulation for random plane partitions and TASEP with arbitrary boundary conditions, enabling all-order asymptotic expansions of plane partition statistics. It establishes a direct link between the symplectic invariants of the mirror spectral curve and Gromov-Witten invariants of C³ with branes, yielding universal regimes and exact solutions via matrix model techniques.

ABSTRACT

We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a (T)ASEP with arbitrary boundary conditions. Using the known solution of matrix models, this method allows to find the large size asymptotic expansion of plane partitions, to ALL orders. It also allows to describe several universal regimes. On the algebraic geometry point of view, this gives the Gromov-Witten invariants of C³ with branes, i.e. the topological vertex, in terms of the symplectic invariants of the mirror’s spectral curve.

Motivation & Objective

• To construct an exact matrix model equivalent to the random plane partition model under almost arbitrary boundary conditions.
• To extend this equivalence to the totally asymmetric exclusion process (TASEP) with arbitrary boundary conditions.
• To derive the full asymptotic expansion of plane partitions to all orders in the large size limit.
• To identify and describe universal scaling regimes in the model.
• To connect the symplectic invariants of the mirror spectral curve to Gromov-Witten invariants of C³ with branes (i.e., the topological vertex).

Proposed method

• Formulate a matrix model that exactly reproduces the partition function of the random plane partition model with general boundary conditions.
• Leverage known exact solution techniques for matrix models to compute all-order asymptotic expansions.
• Map the spectral curve of the matrix model to the mirror geometry of the topological vertex computation.
• Use the Eynard-Orantin topological recursion framework to compute symplectic invariants from the spectral curve.
• Apply the matrix model solution to extract universal scaling limits in the TASEP and plane partition settings.
• Establish a precise correspondence between the symplectic invariants of the mirror curve and the Gromov-Witten invariants of C³ with branes.

Experimental results

Research questions

• RQ1 How can a matrix model be constructed to exactly reproduce the random plane partition model with arbitrary boundary conditions?
• RQ2 What is the all-order asymptotic expansion of the plane partition partition function in the large size limit?
• RQ3 How do universal scaling regimes emerge in the TASEP and plane partition models under general boundary conditions?
• RQ4 What is the precise mathematical relationship between the symplectic invariants of the mirror spectral curve and the Gromov-Witten invariants of C³ with branes?
• RQ5 Can the topological vertex in algebraic geometry be fully reconstructed from matrix model data via spectral curve invariants?

Key findings

• An exact matrix model is constructed that reproduces the random plane partition model with arbitrary boundary conditions, valid beyond asymptotic approximations.
• The method yields the full asymptotic expansion of plane partition statistics to all orders in the large size limit.
• Several universal scaling regimes are identified and described through the matrix model framework.
• The symplectic invariants of the mirror spectral curve are shown to directly compute the Gromov-Witten invariants of C³ with branes.
• The topological vertex, or equivalently the Gromov-Witten invariants of C³ with branes, is expressed in terms of symplectic invariants of the mirror curve.
• The construction establishes a precise and exact correspondence between matrix model solutions and enumerative invariants in algebraic geometry.