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

We show that the vacuum partition function of the q-deformed oscillator \[ L_q(z):= 0|ezT_q|0, T_q := a_q + a_q^* \] satisfies the Lee-Yang property, i.e., all zeros of Lq lie on the imaginary line for all q∈[-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 Lq by expressing φq(z):= Lq(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 φq is real-rooted, and in fact belongs to the Laguerre-Pólya class. The real-rootedness of the q-deformed polynomials yields novel log-concavity relations on the q-deformed matching moments \[M_(q):= ∑π_2(2)qcr(π).\] 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.