Introduction to Phase Transitions in Random Optimization Problems

Reaction-diffusion systems driven far from thermodynamic equilibrium through the injection of energy can support multiple spatial patterns that persist as long-lived dynamical phases. The stability of these metastable phases is not determined by thermodynamics, but by the transition paths connecting them. At finite particle numbers, intrinsic stochasticity induces rare transitions between competing patterns, rendering continuum mean-field descriptions insufficient, while exact stochastic simulations become computationally prohibitive in spatially extended systems. Here, we make absolute instanton rate calculations, including fluctuation and symmetry prefactors, tractable for transitions between metastable patterns in large nonequilibrium reaction-diffusion systems. Applied to a canonical Schlögl model and an experimentally motivated ATP-driven membrane kinase-phosphatase network, the resulting rates agree closely with reaction-diffusion master-equation benchmarks. We show that an effective path entropy can qualitatively alter stability at finite particle numbers by selectively enhancing one switching direction. These results establish path entropy as an organizing principle for finite-particle nonequilibrium pattern stability.

cond-mat.stat-mech↗

arXiv2026

Duality between the level statistics of Hermitian and non-Hermitian random matrices

The complex spectral statistics of non-Hermitian random matrices provide a standard diagnostic of dissipative quantum chaos. Yet an analytical understanding of how symmetry enriches these universal correlations remains limited. We establish an exact duality between the spectral statistics of Hermitian and non-Hermitian random matrices in the limit of large matrix size. This duality resolves a longstanding analytical problem by yielding the previously unknown eigenvalue pair-correlation functions for complex symmetric (class AI$^\dagger$) and complex self-dual matrices (class AII$^\dagger$), completing the analytical description of bulk pair correlations in the non-Hermitian threefold way. It further gives closed-form spectral densities near the origin for seven additional symmetry classes. Numerical tests in interacting and dissipative quantum models support the universality of these predictions beyond Gaussian ensembles. The duality suggests a deeper connection between chaos in closed and open quantum systems.

cond-mat.stat-mech↗

arXiv2026

Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states

Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.

cond-mat.stat-mech↗