In my talk, I will present how to 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 λ 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 λ1>λc1, thus exhibiting a phase transition in λ. 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. This talk is based on arXiv:2204.09998. 

　　报告人简介：吴双，本科及研究生毕业于巴黎第六大学。博士毕业于巴黎萨克雷大学，博士阶段研究的课题为：Algebraic area distribution of two-dimensional random walks and the Hofstadter model。 

　　在上海交通大学物理与天文学院担任博士后，研究课题：利用组合数学的方法研究SYK及其相关的模型。