We compute analytically the probability S(t) that a set of N Brownian paths do not cross each other and stay below a moving boundary g(τ ) = W √ τ up to time t. We show that for large t it decays as a power law S(t) ∼ t −β(N,W ) . The decay exponent β(N, W ) is obtained as the ground state energy of a quantum system of N non interacting fermions in a harmonic well in the presence of an infinite hard wall at position W . Explicit expressions for β(N, W ) are obtained in various limits of N and W , in particular for large N and large W . We obtain the joint distribution of the positions of the walkers in the presence of the moving barrier g(τ ) = W √ τ at large time. We extend our results to the case of N Dyson Brownian motions (corresponding to the Gaussian Unitary Ensemble) in the presence of the same moving boundary g(τ ) = W √ τ . For W = 0 we show that the system provides a realization of a Laguerre biorthogonal ensemble in random matrix theory. We obtain explicitly the average density near the barrier, as well as in the bulk far away from the barrier. Finally we apply our results to N non-crossing Brownian bridges on the interval [0, T ] under a time-dependent barrier gB(τ ) = W τ (1 − τ T ).

The random variable 1 + z 1 + z 1 z 2 + … appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to N × N matrices either real symmetric β = 1 or complex Hermitian β = 2. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to N fermions in a Morse potential, which are non-interacting for β = 2. At finite N the distribution of eigenvalues exhibits heavy tails, generalizing Kesten’s results in the scalar case. The density of fermions in this potential is studied for large N, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large N limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valkó, Grabsch and Texier, as well as Ossipov, is discussed.

We study the quench dynamics in free fermionic systems in the prototypical bipartitioning protocol obtained by joining two semi-infinite subsystems prepared in different states, aiming at understanding the interplay between quantum coherences in space in the initial state and transport properties. Our findings reveal that, under reasonable assumptions, the more correlated the initial state, the slower the transport is. Such statement is first discussed at qualitative level, and then made quantitative by introducing proper measures of correlations and transport ‘speed’. Moreover, it is supported for fermions on a lattice by an exact solution starting from specific initial conditions, and in the continuous case by the explicit solution for a wider class of physically relevant initial states. In particular, for this class of states, we identify a function, that we dub transition map, which takes the value of the stationary current as input and gives the value of correlation as output, in a protocol-independent way. As an aside technical result, in the discrete case, we give an expression of the full counting statistics in terms of a continuous kernel for a general correlated domain wall initial state, thus extending the recent results in Moriya et al (2019 J. Stat. Mech. 2019 063105) on the one-dimensional XX spin chain.

A conditioned stochastic process can display a very different behavior from the unconditioned process. In particular, a conditioned process can exhibit non-Gaussian fluctuations even if the unconditioned process is Gaussian. In this work, we revisit the Ferrari–Spohn model of a Brownian bridge conditioned to avoid a moving wall, which pushes the system into a large-deviation regime. We extend this model to an arbitrary number N of non-crossing Brownian bridges. We obtain the joint distribution of the distances of the Brownian particles from the wall at an intermediate time in the form of the determinant of an N × N matrix whose entries are given in terms of the Airy function. We show that this distribution coincides with that of the positions of N spinless noninteracting fermions trapped by a linear potential with a hard wall. We then explore the N ≫ 1 behavior of the system. For simplicity we focus on the case where the wall’s position is given by a semicircle as a function of time, but we expect our results to be valid for any concave wall function.