q-Random Matrix Ensembles

Abstract

Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matrix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.

Soft Matrix Models and Chern–simons Partition Functions

“…This feature has already been discussed in the context of random matrix theory and also in the particular case of q-deformed matrix models. 16 In this last case, the one we are mainly interested, the non-uniqueness is strong since the potentials are asymptotically V (x) ∼ log 2 x for 16 x → ∞ and we have already mentioned that we are in the soft regime when the confinement provided by the potential is weaker than V (x) ∼ x.…”

The spectral form factor in the `t Hooft limit -- Intermediacy versus universality

“…Whilst, in [1], we found that taking only N Ñ 8 leads to a universal connected SFF in the form of a linear ramp of unit slope, we see here that we get quite different behavior as we take the 't Hooft limit. Further, it is known that the intermediate matrix ensemble is a member of a family of ensembles exhibiting so called weak or soft confinement [49], which correspond to indeterminate moment problems and which have been argued to form their own one-parameter universality class [19], the parameter in question being q. In [1], we found that all invariant one-matrix models arising in topological string theory that are known (to us) belong to this family of soft confining ensembles.…”

“…In section 3.3, we calculate the SFF in the 't Hooft limit and find that the connected SFF reduces to a sequence of polynomials which do not seem to have appeared in the literature thus far. We argue that this could constitute a novel RMT universality, different from the q-dependent universality described in [19]. Then, in section 3.4, we explore the non-commutativity of the limits N Ñ 8 and q Ñ 1 and calculate the SFF for small 't Hooft parameter.…”

“…In the latter limit, which is essentially the 't Hooft limit, an approximate expression for the parametric density correlation function was found in [41]. Further, a similar limit was considered for another, closely related, q-deformed circular unitary ensemble in [42], see also [19]. It was found that deviations from the CUE level density only persist in the infinite N limit if one simultaneously scales q such that p1 ´qqN remains finite, which is essentially the 't Hooft limit.…”

Schur expansion of random-matrix reproducing kernels

“…Throughout the present section we compute the coefficients in the Schur expansion of the kernel for a selected class of q-ensembles. The definition of q-ensemble is the one put forward in [65], namely, a standard random matrix ensemble whose weight function is such that its associated orthogonal polynomials are q-deformed. We will focus on the Stieltjes-Wigert ensemble, which has multitude of physical applications [26,77,78,70,29], and its one-parameter generalization, the q-Laguerre ensemble, which of course is also a one-parameter generalization of the Laguerre ensemble studied above.…”