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

In this paper, we study the asymptotic behaviour of plane partitions distributed according to a $q^{\text{Volume}}$-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.