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

TL;DR

This work tackles the universal ramp in the spectral form factor (SFF) of the double-scaled SYK model by building a chord-diagram framework that rewrites double-trace correlators as sums over single-trace operator insertions. In the $q=0$ random-matrix theory limit, non-crossing chord diagrams reproduce the known local eigenvalue correlations via a transfer-matrix formalism tied to $q$-Hermite polynomials, yielding the Brezin-Zee universal two-point function and a trumpet/gluing interpretation of the cylinder partition function. Extending to finite $q$ introduces allowed chord intersections and reveals a finite-$q$ late-time ramp characterized by a $q$-deformed trumpet function $F_{eta,k}(q)$ and a topological gluing factor $C_k$, with the ramp asymptotics matching the universal form $G_{late}(eta,t) \propto t$ at late times and edge energy $E_0=2/\sqrt{1-q}$. Overall, the paper provides a concrete, diagrammatic route to the RMT ramp within DSSYK and proposes a chord-based analog to topological recursion that interpolates between pure RMT and finite-$q$ DSSYK, offering insights into the gravitational interpretation of these spectral fluctuations.

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.

Figures (11)

• Figure 1: a Chord diagram on the left with 8 nodes representing 8 copies of the Hamiltonian. Each node has $p$-interacting fermions with associated random coupling. On the right, we open this chord diagram into a straight line.
• Figure 2: The Chord diagram interpretation of $m_4$ amounts to 3 configurations: two of them with no intersections and the last one with one intersection $=2+q$, the same value we are getting from wick contraction.
• Figure 3: The SFF of the sparse SYK model averaged over 100 realizations, with $N=30$, $p=4$, $\beta = 1$, and sparseness parameter $k=8$, based on data from C_ceres_2022. The three distinct timescales are separated by dotted vertical lines, and labeled.
• Figure 4: $F_{n,k}$ counts every possible configuration of planar diagrams of a single boundary with $n$ Hamiltonian insertions, $k$ of which connect to the other trace.
• Figure 5: A contribution to the leading order connected double-trace moments in RMT where some of the Hamiltonian chords (blue) cross the marked chord (red), but no crossings exist between Hamiltonian chords. One can also view these diagrams as non-crossing chord diagrams in the topology of a disk with a single handle-body. Due to the crossing, the marked chord must be the identity operator to ensure a non-vanishing contribution.