Sparse Sachdev-Ye-Kitaev model, quantum chaos, and gravity duals

We study a sparse Sachdev-Ye-Kitaev (SYK) model with $N$ Majoranas where only $\ensuremath{\sim}kN$ independent matrix elements are nonzero. identify minimum $k\ensuremath{\gtrsim}1$ for quantum chaos to occur by level statistics analysis. The spectral density in this region, and larger $k$, is still given the Schwarzian prediction of dense SYK model, though renormalized parameters. Similar results obtained beyond linear scaling number nonzero elements. This strong indication that connectivity have gravity dual. also find an intriguing exact relation between leading correction moments due sparsity $1/d$ Parisi's U(1) lattice gauge theory $d$-dimensional hypercube. In $k\ensuremath{\rightarrow}1$ limit, different disorder realizations show emergent random fixed can be any universality class tenfold way. agreement restricted short-range correlations, no more than few spacings, particular tail spectrum. addition, discrete global symmetries most $k$ slightly below one give rise ${2}^{m}$-fold degenerate spectra, $m$ being positive integer. For $k=3/4$, we observe large such maximum ${2}^{8}$-fold spectra $N=26$.