#### Volume 20, issue 6 (2020)

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Motivic random variables and representation stability, I: Configuration spaces

### Sean Howe

Algebraic & Geometric Topology 20 (2020) 3013–3045
DOI: 10.2140/agt.2020.20.3013
##### Abstract

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $ℂ$ 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.

##### Keywords
representation stability, motivic stabilization, arithmetic statistics, configuration spaces, cohomological stability.
##### Mathematical Subject Classification 2010
Primary: 14G10, 18F30, 55R80