The statistical fluctuations of transport quantities, such as conductance, can serve as powerful tools for probing the underlying quantum nature of a system. The most well-known example of this is the phenomena of universal conductance fluctuations (UCF), where the conductance fluctuations in weakly-interacting disordered metallic samples at zero temperature exhibit \(\mathcal{O}(1)\) fluctuations (in units of the conductance quanta, \(e^2 / h\) ) upon tuning external parameters such as chemical potential and magnetic field. Remarkably, the magnitude of these fluctuations in certain limits are independent of microscopic details and are only sensitive to properties like dimensionality and symmetry. A useful overview of UCF is given here, with a more technical review here.

In this work, we analyze how fluctuations in transport quantities manifest in strongly-interacting systems, and how this can be used as an experimental probe of correlated physics. Specifically, we study an interacting dot modeled by a complex SYK interaction - with random four-fermion interactions - along with a random single-particle hopping. The quantum dot is coupled to two leads; we assume the coupling to the leads is the smallest energy scale in the problem, such that to leading order the transport properties can be determined by an analysis of the isolated dot.

The average values of transport properties have been studied before - the conclusion is that there exists a coherence energy \(E_{coh} = t^2 / J \), such that Fermi liquid behavior appears for low temperatures \( T \ll E_{coh} \) and SYK for \( T \gg E_{coh} \). This crossover behavior follows from the self-consistent solution of the Schwinger-Dyson equations, which is exact in the large-\(N\) limit. One of our main results is that this general picture does not hold true for higher moments of transport quantities.