# Multi-time distribution in discrete polynuclear growth

## TL;DR

This paper derives a formula for the joint multi-time distribution in a discrete polynuclear growth model, using contour integrals and Fredholm determinants, and computes its asymptotic form under KPZ scaling, highlighting universality in the KPZ class.

## Contribution

It provides the first explicit formula for multi-time distributions in discrete polynuclear growth models and connects it to the KPZ universality class through asymptotic analysis.

## Key findings

- Derived a multiple contour integral formula for multi-time distribution
- Computed the asymptotic distribution under KPZ scaling limit
- Confirmed universality of the distribution in the KPZ class

## Abstract

We study the multi-time distribution in a discrete polynuclear growth model or, equivalently, in directed last-passage percolation with geometric weights. A formula for the joint multi-time distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multi-time distribution is then computed by taking the appropriate KPZ-scaling limit of this formula. This distribution is expected to be universal for models in the Kardar-Parisi-Zhang universality class.