A new study proves that the vacuum partition function of a q-deformed oscillator, L_q(z) = ⟨0|e^{zT_q}|0⟩ with T_q = a_q + a_q*, satisfies the Lee-Yang property: all its zeros lie on the imaginary axis for every q ∈ [-1, 1). The result settles conjectures raised by Okuyama in 2023 and connects to the Sachdev-Ye-Kitaev (SYK) model in the doubly-scaled regime, where the annealed partition function also satisfies the Lee-Yang property in the thermodynamic limit.

The starting point is the observation that q-deformed oscillators are related to the SYK Hamiltonian in the doubly-scaled limit, a version of the random-fermion model that has gained prominence in studies of quantum gravity and black holes. Through this connection, the authors transfer the result on the zeros of L_q to the SYK case: all Fisher zeros of the annealed partition function lie on the imaginary axis. As a consequence, the quenched partition function admits a zero-free strip of constant width with high probability, rigorously complementing the large-k replica calculations of Zlokapa and Kiani.

The proof rests on writing φ_q(z) = L_q(iz) as a locally uniform limit of q-deformed matching polynomials and showing that those polynomials are real-rooted. This places φ_q in the Laguerre-Pólya class, guaranteeing that its zeros are imaginary. The real-rootedness of the q-deformed polynomials is established via the Touchard-Riordan formula, which yields a partial differential equation relating q-derivatives to z-derivatives and allows the motion of the roots to be analyzed.

The result also yields novel log-concavity relations for the q-deformed matching moments M_ℓ(q) = Σ_{π∈P₂(2ℓ)} q^{cr(π)}, where cr(π) counts the crossings of a pairing partition. The work opens the door to applying the Lee-Yang property to other random-fermion models and to exploring further the link between real-rootedness of combinatorial polynomials and phase-transition phenomena in disordered quantum systems.