Non-crossing Brownian Paths and Dyson Brownian Motion Under a Moving Boundary

Abstract

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 ).

Riemann surfaces for KPZ with periodic boundaries

“…where b j (x)−b j (0), j ∈ Z are independent standard Brownian bridges with b j (0) = b j (1), distances between consecutive endpoints b j+1 (0) − b j (0) are independent exponentially distributed random variables with parameter s, and b 0 (0) = h 0 (0). Deriving exact formulas from the Brownian bridge representation is still an open problem, though, even for the simple initial conditions (flat, Brownian, sharp wedge) for which the result is known from Bethe ansatz, see however [48] for related work. The idea that the contributions of the excited states of a theory should follow from that of the ground state by analytic continuation with respect to some parameter is not new, see for instance [49] for the quantum quartic oscillator, [50] for the Ising field theory on a circle (where the ground state energy is interestingly also given by an infinite sum of square roots, but with conjugate branch points paired), or [51] for models described by the thermodynamic Bethe ansatz.…”

Interplay between transport and quantum coherences in free fermionic systems

“…In a larger scope, we note that connections with random matrix theory and stochastic systems also exist for other fermionic systems, namely non-interacting fermions in a range of classic potentials, see for instance Ref. [52][53][54].…”

A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

“…[32,40,59]. We also refer to [11,10] and [35] for the emergences of Muttalib-Borodin ensemble in plane partitions and the Dyson Brownian motion under a moving boundary.…”