Lee-Yang theorems for $q$-deformed oscillators and doubly-scaled SYK

Abstract

We show that the vacuum partition function of the $q$-deformed oscillator \[ L_q(z):= \langle0|e^{zT_q}|0\rangle, \qquad T_q := \mathsf{a}_q + \mathsf{a}_q^* \] satisfies the Lee-Yang property, i.e., all zeros of $L_q$ lie on the imaginary line for all $q\in[-1, 1)$, proving the conjectures of Okuyama'23. Using the connection between $q$-deformed oscillators and the Sachdev-Ye-Kitaev (SYK) Hamiltonian in the doubly-scaled regime, we show that the annealed SYK partition function satisfies the Lee-Yang property in the thermodynamic limit, that is, all of its Fisher zeros are on the imaginary axis. As a result, we show that the quenched SYK partition function admits a zero-free strip of constant width with high probability, rigorously complementing the large$-k$ replica calculations of Zlokapa and Kiani'26. We show the imaginary-rootedness of $L_q$ by expressing $φ_q(z):= L_q(\mathbf{i} z)$ as a locally uniform limit of ``$q$-deformed'' matching polynomials and then showing that those $q$-deformed polynomials are real-rooted. This establishes that $φ_q$ is real-rooted, and in fact belongs to the \emph{Laguerre-Pólya} class. The real-rootedness of the $q$-deformed polynomials yields novel log-concavity relations on the \emph{$q$-deformed matching moments} \[M_\ell(q):= \sum_{π\in\mathcal{P}_2(2\ell)}q^{\operatorname{cr}(π)}.\] We show the real-rootedness of the $q$-deformed polynomials by using the Touchard--Riordan formula to derive a partial differential equation relating their $q$-derivatives to their $z$-derivatives, and then analyzing the resulting motion of their roots.