The late time ramp from chord diagrams in the double-scaled SYK model

Abstract

We compute the ramp of the spectral form factor analytically from chord diagrams in double scaled SYK. We map the double-trace correlator to a sum of single trace two-point functions over a basis of operators. We then reproduce the local eigenvalue correlations in random matrix theory from the chord diagrams perspective, which is the $q= 0$ limit of double scaled SYK, and identify the relevant operators that give rise to the late-time ramp. We then extend the computation to finite $q$, resulting in the late time contribution to the spectral form factor. We verify that the late time asymptotics of the finite $q$ computation gives rise to the expected late time ramp. Our computation also provides the corresponding trumpet partition function and gluing factor for chords, which form the basis of a chord analog to topological recursion.