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Does the heat flow conjecture for random matrices extend to infinite-dimensional operator algebras, and if so, what is the correct formulation of the limiting zero distribution?

Related: Circular Law Theorem, Free Entropy conjecture of Voiculescu, Wigner semicircle law

Solving this problem would unlock a unified framework connecting infinite-dimensional stochastic processes with free probability and random operator theory, potentially providing new tools for quantum field theory where infinite-dimensional random operators appear naturally as fields. It would also clarify whether universality phenomena seen in finite random matrix theory, such as edge statistics and bulk correlations, persist in some form for operator-valued stochastic processes, with implications for the study of stochastic partial differential equations and their spectral properties.

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