Motivic random variables and representation stability I: Configuration spaces

TL;DR

This paper establishes a motivic stabilization phenomenon for the cohomology of configuration spaces over complex varieties, interpreting it through probabilistic independence of motivic random variables and providing explicit formulas for the limits.

Contribution

It introduces a new probabilistic approach to motivic stabilization in the cohomology of configuration spaces, with explicit universal formulas and potential applications to other geometric spaces.

Findings

Proves motivic stabilization for cohomology of configuration spaces

Provides explicit formulas for the limits using motivic Euler products

Establishes a probabilistic interpretation of the stabilization phenomenon

Abstract

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In the sequel we find similar stability results in spaces of smooth hypersurface sections, providing new examples to be investigated through the lens of representation stability for symmetric, symplectic and orthogonal groups.