Amari Jaconelli

Finite Temperature GUE Kernels, AKNS, and All That

We investigate a finite-temperature generalisation of the finite-N GUE kernel and derive connections with integrable systems for the corresponding Fredholm determinant. In the most general case, we obtain a non-Abelian, integro-differential version of the AKNS system. Furthermore, for a specific specialisation, we find an integro-differential version of the Painlevé IV sigma form and a non-Abelian version of the first and second moments of the Merola-Ragnisco-Tu lattice system. The former can be understood as a generalisation of the classical Tracy-Widom result that links the finite-N GUE kernel to the sigma-Painlevé IV equation.