Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization

Abstract

We revisit the problem of an elastic line (such as a vortex line in a superconductor) subject to both columnar disorder and point disorder in dimension d = 1 + 1. Upon applying a transverse field, a delocalization transition is expected, beyond which the line is tilted macroscopically. We investigate this transition in the fixed tilt angle ensemble and within a "one-way" model where backward jumps are neglected. From recent results about directed polymers in the mathematics literature, and their connections to random matrix theory, we find that for a single line and a single strong defect this transition in presence of point disorder coincides with the Baik-Ben Arous-Péché (BBP) transition for the appearance of outliers in the spectrum of a perturbed random matrix in the Gaussian Unitary Ensemble. This transition is conveniently described in the polymer picture by a variational calculation. In the delocalized phase, the ground state energy exhibits Tracy-Widom fluctuations. In the localized phase we show, using the variational calculation, that the fluctuations of the occupation length along the columnar defect are described by fKPZ, a distribution which appears ubiquitously in the Kardar-Parisi-Zhang universality class. We then consider a smooth density of columnar defect energies. Depending on how this density vanishes at its lower edge we find either (i) a delocalized phase only (ii) a localized phase with a delocalization transition. We analyze this transition which is an infinite-rank extension of the BBP transition. The fluctuations of the ground state energy of a single elastic line in the localized phase (for fixed columnar defect energies) are described by a Fredholm determinant based on a new kernel, closely related to the kernel describing the largest real eigenvalues of the real Ginibre ensemble. The case of many columns and many non-intersecting lines, relevant for the study of the Bose glass phase, is also analyzed. The ground state energy is obtained using free probability and the Burgers equation. Connections with recent results on the generalized Rosenzweig-Porter model suggest that the localization of many polymers occurs gradually upon increasing their lengths.

The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall

“…where the CDF F BBP a (s) was introduced in [31] and governs the fluctuations of the eigenvalues of spiked Hermitian matrices. It was also found to arise in the context of the KPZ universality class in full-space models for half-Brownian type IC [14,15,36] and in other contexts [31,[37][38][39]. In particular, for a = 0, 2 where F 1 is the Tracy-Widom distribution function for the largest eigenvalue of a GOE random matrix.…”

“…In this Section, we explain the identity in law (39) (which in particular implies the identity in distribution (37) after letting B go to +∞). Recall the definition of Z (B|A,B) (X, t), that is the solution to full-line SHE (34) with IC depending on parameters A, B and specified by (40) and (41).…”

“…For this, we first need to establish a moment formula for certain partition functions of the log-gamma polymer, a directed polymer model on the lattice Z 2 introduced in [57]. Then we will take the continuous limit to the KPZ equation along the lines of [29], and find that the moments match with (76), up to a factor 2 that accounts for the factor 2 in (39).…”

“…Our asymptotic results at large time for the KPZ equation in Section II thus confirm universality predictions. Let us stress, however, that the identity in distribution from [20] cannot be scaled to the KPZ equation: one cannot deduce from it our finite time identities in distribution (35), (37), (39) and (87).…”

Non-intersecting Brownian bridges in the flat-to-flat geometry

“…It is then easy to show that, in the large N limit, this Green's function also satisfies the inviscid Burgers' equation [37][38][39][40]…”