Discrete and Continuous Muttalib--Borodin Process: Large Deviations and Limit Shape Analysis

2025/05/29 by Jonathan Husson, Husson, Jonathan, Guido Mazzuca +3
Mathematics · #34M50 #60F10 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2505.23164

openalex publication_date 2025/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the asymptotic behaviour of plane partitions distributed according to a q^Volume-weighted Muttalib--Borodin ensemble and its associated discrete point process. We establish a Large Deviation Principle for the process, explicitly characterizing the rate function. A defining feature of our model is the emergence of a strict upper bound on the macroscopic particle density, which translates the asymptotic analysis into a non-trivial constrained minimization problem. Through a rigorous Riemann--Hilbert analysis, we derive exact, closed-form formulas for the limit shape of the partitions across all parameter regimes. To the best of our knowledge, this represents the first time a constrained Riemann--Hilbert problem has been formulated and analytically solved for a bi-orthogonal ensemble. Our analysis allows to track the system through a macroscopic phase transition, computing the minimizer in both the subcritical and supercritical regimes. As a byproduct of our analysis, we obtain an explicit expression for the arctic curve that separates the ``frozen'' and ``liquid'' regions of the limit shape. Furthermore, we reveal that the equilibrium measure exhibits a continuously varying exponent at the hard edge departing from the universal fixed exponents typically observed in classical random matrix theory.

Citations

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Related

• Discrete and continuous Muttalib--Borodin processes I: the hard edge
• Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles
• A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles
• Higher order large gap asymptotics at the hard edge for Muttalib--Borodin ensembles
• Hard edge universality of Muttalib-Borodin ensembles with real parameter θ