Thermodynamic limit and dispersive regularization in matrix models

Abstract

We show that Hermitian matrix models support the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a solution of a reduction of the Toda hierarchy, known as Volterra system, we argue that the singularity is resolved by the onset of a multi-dimensional dispersive shock of the order parameter in the space of coupling constants. This analysis explains the origin and mechanism leading to the emergence of chaotic behaviours observed in M 6 matrix models and extends its validity to even nonlinearity of arbitrary order.

Symmetric matrix ensemble and integrable hydrodynamic chains

“…The system (B.1) and its relation with the large n scaling properties of the initial condition will be analysed in detail in a separate work. The previously unseen connection between matrix ensembles and hierarchies of hydrodynamic chains discussed in this paper, together with the aforementioned results for the HME discussed in [8], suggests that the study of random matrix models may lead to the discovery of new interesting integrable hydrodynamic PDEs. The study of the PDEs so obtained arises as a general framework and a new methodology to classify and describe asymptotic properties of complex systems.…”

“…It is indeed well known that, for generic initial conditions, solutions of systems of hydrodynamic type break down in finite time, namely, in the context of SME, for finite values of the coupling constants t k . In the case of the HME, the critical behaviour of the order parameter at the leading order is described by the Whitney cusp and, as observed in [8], finite size corrections resolve the singularity via the onset of a modulated highly oscillating quasi-periodic wave, known as dispersive shock. The dispersive shock characterises a new type of phase transition where asymptotic stable states are connected by an intermediate state where order parameters develop fast oscillations induced by the dispersive nature of finite size corrections.…”

“…For even power interactions, the thermodynamic limit of the HME is constituted by an order parameter that evolves according to a scalar integrable hierarchy (the Hopf hierarchy) [8]. On the other hand, in the n → ∞ limit, the even SME is specified by infinitely many order parameters that satisfy an integrable hydrodynamic chain.…”

“…The derivation described above naturally compares with the case of the HME, as studied in [8], where the partition function corresponds to a particular solution of the Toda Lattice and the reduction of even power interactions provides a solution to the Volterra Lattice. The Volterra Lattice is effectively an independent integrable system as it arises from a reduction in the even Toda hierarchy and it is not compatible with the odd flows of the Toda hierarchy.…”

“…In the present paper, we propose an approach to the study of the Symmetric Matrix Ensemble (SME) based on the method of differential identities developed and effectively applied to a variety of statistical mechanical models (see, e.g. [7,8,21,23]). In the case of SMEs, the method relies on its underlying integrable structure realised by the Pfaff Lattice [5,6,17,31,32].…”