# Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary

## TL;DR

This paper analytically studies the probability that multiple Brownian paths and Dyson Brownian motions stay below a moving boundary, revealing power-law decay and connections to random matrix theory, with explicit results for large systems and specific boundary conditions.

## Contribution

It provides explicit formulas for the non-crossing probability decay exponent and extends results to Dyson Brownian motions and non-crossing bridges with moving boundaries.

## Key findings

- Decay exponent $eta(N,W)$ derived as quantum ground state energy.
- Explicit expressions for $eta(N,W)$ in various limits.
- Connection to Laguerre biorthogonal ensemble in random matrix theory.

## 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 $g(\tau)= W \sqrt{\tau}$ up to time $t$. We show that for large $t$ it decays as a power law $S(t) \sim t^{- \beta(N,W)}$. The decay exponent $\beta(N,W)$ 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 $\beta(N,W)$ 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 $g(\tau) =W \sqrt{\tau}$ 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 $g(\tau)=W\sqrt{\tau}$. 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 $g_B(\tau)= W \sqrt{\tau(1- \frac{\tau}{T})}$.