Order parameters for unitary ensembles solve the modified KP equation via the Volterra lattice while orthogonal ensembles yield a new integrable chain via the Pfaff lattice, with thermodynamic limits being hydrodynamic-type systems solved by the same semi-discrete dynamical chain.
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Partial results on low-lying zero densities imply explicit conditional lower bounds on central L-values, with bound quality tied to family symmetry type and allowed Fourier support.
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Random matrix ensembles and integrable differential identities
Order parameters for unitary ensembles solve the modified KP equation via the Volterra lattice while orthogonal ensembles yield a new integrable chain via the Pfaff lattice, with thermodynamic limits being hydrodynamic-type systems solved by the same semi-discrete dynamical chain.
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A connection between low-lying zeros and central values of $L$-functions
Partial results on low-lying zero densities imply explicit conditional lower bounds on central L-values, with bound quality tied to family symmetry type and allowed Fourier support.