Quenched properties of the spectral form factor

The Spectral Form Factor (SFF) is defined for Hermitian matrices as the modulus squared of the partition function in complex temperature, with a suitable generalization existing in the non-Hermitian case. In this work, we study the properties of what we refer to as the quenched SFF, namely the logarithm of the SFF and in particular its average, and compare them with those of its ordinary (annealed) counterpart, namely the average of the SFF (and eventually its logarithm). While the SFF is famously not self-averaging, the opposite is true for the quenched SFF, in both the Hermitian and non-Hermitian cases. Nonetheless, the quenched and the annealed averages coincide up to subleading constants, at least for high enough temperatures. The fluctuations of lnSFF are deep and one encounters thin spikes when moving close to a zero of the partition function. In order to study the fluctuations of the quenched SFF at late times we consider a suitable change of variable of lnSFF which turns out to be compatible with a Gumbel distribution. We note that the exponential tail of this distribution can indeed be obtained by sampling the deep spikes of lnSFF, namely the Fisher zeros of the partition function. We compare with the results obtained in isolated many-body systems and we show that same results hold at late times also for non-Hermitian Hamiltonians and non-Hermitian random matrices.