Lee-Yang theorems for q-deformed oscillators and doubly-scaled SYK
Abstract
We show that the vacuum partition function of the q-deformed oscillator Lq(z):= ⟨0|ezTq|0⟩, Tq := aq + aq* 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 \emph{Laguerre-Pólya} class. The real-rootedness of the q-deformed polynomials yields novel log-concavity relations on the \emph{q-deformed matching moments} Mℓ(q):= ∑π∈𝒫2(2ℓ)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.
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BibTeX
@misc{basu2026leeyang,
title = {{Lee-Yang theorems for $q$-deformed oscillators and doubly-scaled SYK}},
author = {Arpon Basu and Siddhant Midha},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03624},
eprint = {2611.03624},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03624},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lee-Yang theorems for q-deformed oscillators and doubly-scaled SYK
AU - Arpon Basu
AU - Siddhant Midha
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03624
UR - https://arxiv.org/abs/2611.03624
AB - We show that the vacuum partition function of the q-deformed oscillator L_q(z):= ⟨0|e^zT_q|0⟩, T_q := a_q + a_q^* satisfies the Lee-Yang property, i.e., all zeros of L_q 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 L_q by expressing φ_q(z):= L_q(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_ℓ(q):= ∑_π∈𝒫_2(2ℓ)q^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.
ER -