# Boundary Crossing in Stochastic Inflation with Critical Number of Fields

## TL;DR

This paper investigates how the probability of the inflaton crossing boundaries in stochastic inflation depends on the number of fields, revealing a critical threshold at two fields, with exact calculations for specific multi-well potentials.

## Contribution

It demonstrates the critical dependence of boundary crossing probability on the number of fields in multi-field inflation and provides exact calculations for certain models.

## Key findings

- Boundary crossing probability is nonzero for more than two fields.
- Probability is zero for two or fewer fields.
- Exact solutions are provided for specific multi-well potentials.

## Abstract

We study boundary crossing probability in the context of stochastic inflation. We prove that for a generic multi-field inflationary potential, the probability that the inflaton reaches infinitely far regions in the field space is critically dependent on the number of fields, being nonzero for more than two fields, and zero otherwise. We also provide several examples where the boundary crossing probability can be calculated exactly, most notably, for a particular landscape of a two-field model with a multi-well potential.