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.