SYK model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum

We study the SYK model with an extra constant source, .i.e. a constant matrix or equivalently a diagonal matrix with only one non-zero entry ${\lambda }_1$. By using methods from analytic combinatorics, we find exact expressions for the moments of this model. We further prove that the spectrum of this model can have a gap when ${\lambda }_1>{\lambda }_1^c$, thus exhibiting a phase transition in ${\lambda }_1$. In this case, a single isolated eigenvalue splits off from SYK’s eigenvalues distribution. We located this single eigenvalue by analyzing the singular behavior of a supercritical functional composition scheme. In certain limit our results recover the ones of random matrices with non-zero mean entries.