Complex random energy model : zeros and fluctuations

The partition function of the random energy model at inverse temperature $\beta$ is defined by $Z_N(\beta) = \sum_{k=1}^N \exp(\beta \sqrt{n} X_k)$, where $X_1, X_2, \ldots$ are independent real standard normal random variables, and $n = log N$. We identify the asymptotic structure of complex zeros of $Z_N$, as $N \rightarrow \infty$, confirming predictions made in the theoretical physics literature. Also, we describe the limiting complex fluctuations for a model generalizing $Z_N(\beta)$.