Rigorous analytic structure of the double-scaled SYK partition function: high-temperature coefficient asymptotics, zeros of Z(it), and the infinite-product representation

The double-scaled SYK model (DSSYK) has an exactly known partition function $Z(\beta)=\langle 0|e^{\beta T}|0\rangle$, where $T$ is a $q$-deformed oscillator transfer matrix acting on chord states with $q=e^{-\lambda}$. Okuyama (2023) found numerically that the coefficients $f_n(q)$ of the high-temperature expansion of the free energy obey $f_n(q)\sim(1-q)\,n^{-1}A(q)^{-2n}$, with $A(q)$ known only numerically except at the endpoints $q=0$ (where $A=j_{1,1}/2$, half the first Bessel zero) and $q=1$ (where $A=\max_s 2s/\cosh s$), and conjectured that the radius of convergence equals the first zero $t_1(q)=A(q)/\sqrt{1-q}$ of $Z(\mathrm{i}t)$ and, more generally, that $Z(\beta)=\prod_{i\geq 1}\bigl(1+\beta^2/t_i(q)^2\bigr)$ in terms of the zeros $t_i(q)$ of $Z(\mathrm{i}t)$ — a genus-zero product asserting that all zeros lie on the imaginary axis. None of these statements has been proven. The problem asks for (1) a rigorous derivation of the large-$n$ asymptotics together with an exact characterization of $A(q)$, (2) a proof that the zeros of $Z(\mathrm{i}t)$ are real, positive and simple and that the product representation holds, and (3) the limit shape of the $q$-weighted Dyck paths underlying the moments and its relation to the boundary-particle Brownian motion of the proposed bulk dual. The three parts share one objective: a proof-level account of how the chord combinatorics of DSSYK determines the singularity structure of $Z(\beta)$.

Background

The Sachdev–Ye–Kitaev (SYK) model is a quantum-mechanical system of $N$ randomly coupled fermions that is solvable at strong coupling and behaves like a holographic dual of near-extremal black holes. In the double-scaled SYK (DSSYK) limit, both $N$ and the number $p$ of fermions per interaction term grow with $p^2/N$ held fixed, and the model becomes exactly solvable. In this regime the Hilbert space is organized by chords — line segments joining fermion insertions in the large-$p$ diagrammatic expansion — and the chord number plays the role of an excitation number. The thermal partition function is $Z(\beta)=\langle 0|e^{\beta T}|0\rangle$, where the transfer matrix $T=A_-+A_+$ acts on chord-number states through a $q$-deformed harmonic oscillator (a $q$-oscillator: an oscillator whose ladder algebra is deformed by a parameter $q$, here $q=e^{-\lambda}$ with $\lambda\propto p^2/N$), $A_-|n\rangle=\sqrt{(1-q^{n})/(1-q)}\,|n-1\rangle$ and $A_+|n\rangle=\sqrt{(1-q^{n+1})/(1-q)}\,|n+1\rangle$. The spectrum of $T$ is $2\cos\theta/\sqrt{1-q}$ and is bounded, so $Z(\beta)$ is an entire function of the inverse temperature $\beta$. Diagonalizing $T$ with $q$-Hermite polynomials gives the exact integral representation

$$Z(\beta)=\int_0^\pi\frac{d\theta}{2\pi}(q;q)_\infty(e^{2\mathrm{i}\theta};q)_\infty(e^{-2\mathrm{i}\theta};q)_\infty\exp\!\Bigl(\frac{2\beta\cos\theta}{\sqrt{1-q}}\Bigr),$$

where $(a;q)_\infty=\prod_{n\geq 0}(1-aq^n)$ is the $q$-Pochhammer symbol.

Expanding $e^{\beta T}$, the moments $\langle 0|T^{2n}|0\rangle$ enumerate the intersection numbers of chord diagrams and are given exactly by the Touchard–Riordan formula, $\langle 0|T^{2n}|0\rangle=(1-q)^{-n}\sum_{j=-n}^{n}(-1)^j q^{j(j-1)/2}\binom{2n}{n+j}$ — a weighted sum over Dyck paths (lattice paths of $2n$ steps that never fall below the horizontal axis, encoding the chord pairings) with the same weights that appear in the $q$-semicircular law (the $q$-deformed analogue, in free probability — the non-commutative counterpart of classical probability — of the Wigner semicircle law) studied by Josuat-Vergès. Okuyama (2023) analyzed the high-temperature (small-$\beta$) expansion of the free energy,

$$\log Z(\beta)=\sum_{n=1}^{\infty}(-1)^{n-1}(1-q)^{n-1}\beta^{2n}f_n(q),$$

and found, numerically (Richardson extrapolation of the exact Touchard–Riordan data up to $n=100$), the large-order law

$$f_n(q)\sim\frac{1-q}{n}A(q)^{-2n},\qquad n\to\infty,$$

where even the prefactor $c(q)=1-q$ is a numerical finding. The amplitude $A(q)$ is known only numerically, except at the endpoints $q=0$, where $A(0)=j_{1,1}/2$ with $j_{1,1}$ the first positive zero of the Bessel function $J_1$, and $q=1$, where $A(1)=\max_{s\in\mathbb{R}}\,2s/\cosh s$ (a value fixed by the semi-classical saddle-point analysis). At $q=0$ the partition function is exactly $Z(\beta)=I_1(2\beta)/\beta$, so $Z(\mathrm{i}t)=J_1(2t)/t$ and its zeros are the Bessel zeros.

A nearest singularity that is a simple zero on the imaginary axis produces exactly this geometric growth with a $1/n$ prefactor. Motivated by this and by 20-digit numerical agreement at $q=0$, Okuyama conjectured that the radius of convergence of the high-temperature series equals the first zero $t_1(q)$ of $Z(\mathrm{i}t)$ on the positive real $t$-axis, with $t_1(q)=A(q)/\sqrt{1-q}$, and, more generally, stated the expectation that $Z(\beta)$ admits the genus-zero infinite product

$$Z(\beta)=\prod_{i=1}^{\infty}\Bigl(1+\frac{\beta^2}{t_i(q)^2}\Bigr),$$

where $t_i(q)$ is the $i$-th zero of $Z(\beta)$ along the imaginary axis; a genus-zero product means an entire function all of whose zeros lie on the imaginary axis and enter the factorization in conjugate pairs. Later work on the same object gives, on the imaginary axis, the bilateral (two-sided) Bessel series $Z(\mathrm{i}t)=\sum_{m\in\mathbb{Z}}q^{m(m-1)/2}J_{2m}\bigl(2t/\sqrt{1-q}\bigr)$ whose $q$-weights $q^{m(m-1)/2}$ are all positive (since $m(m-1)/2$ is a non-negative integer for every $m\in\mathbb{Z}$; the formula is stated in the $\beta$-normalization of Okuyama (2023), while some later works use a normalization in which the Bessel argument carries an extra factor $1/\sqrt{\lambda}$), so the zero conjecture is a concrete statement about a $q$-deformed Bessel-type entire function; the classical Laguerre–Pólya theory (the standard framework for entire functions with real zeros) does not cover it. The third ingredient is combinatorial: the moments $\langle 0|T^{2n}|0\rangle$ are sums over Dyck paths of $2n$ up/down steps in which each path carries the product of the $q$-oscillator Jacobi coefficients $b_h=\sqrt{(1-q^{h})/(1-q)}$ along the path (the Touchard–Riordan formula above is an equivalent closed form of the same moments, not a separate ensemble), and while the limit shape (the dominant deterministic profile $f(t)$ approached by the height-rescaled weighted paths as $n\to\infty$) is known for the ordinary, non-deformed oscillator from the work of Avdoshkin and Dymarsky on Euclidean operator growth — heights rescaled linearly in $n$ to a profile $f(t)$, $t\in[0,1]$, $f(0)=f(1)=0$, $f\geq 0$, selected as the saddle point of the corresponding path integral — the $q$-deformed case is open; Berkooz, Isachenkov, Narayan and Narovlansky connected chords to quantum groups and to the Brownian motion of a particle on the boundary of a non-commutative $AdS_2$ — a dynamics that emerges in the $q\to 1$ low-energy limit — suggesting that the DSSYK limit shape should reproduce that boundary dynamics.

Problem Statement

Determine, with mathematical rigor, the analytic structure of the DSSYK partition function $Z(\beta)=\langle 0|e^{\beta T}|0\rangle$ at fixed $q\in[0,1)$, where $T=A_-+A_+$ is the $q$-deformed oscillator transfer matrix acting on chord states, together with the combinatorial mechanism behind it. The problem has three linked deliverables.

(1) Large-order asymptotics. Prove that the coefficients $f_n(q)$ defined by $\log Z(\beta)=\sum_{n\geq 1}(-1)^{n-1}(1-q)^{n-1}\beta^{2n}f_n(q)$ satisfy $f_n(q)\sim\frac{1-q}{n}A(q)^{-2n}$ as $n\to\infty$ for each fixed $q\in[0,1)$, and characterize $A(q)$ exactly — by a closed form, an implicit spectral or variational definition, or an equivalent rigorous description — reproducing the endpoint values $A(0)=\frac{1}{2}j_{1,1}$ ($j_{1,1}$ the first positive zero of the Bessel function $J_1$) and $A(1)=\max_{s\in\mathbb{R}}2s/\cosh s$; the $A(1)$ condition is a limit statement, as in Okuyama (2023): $A(q)$ remains finite as $q\to 1^-$ with this limiting value, while $t_1(q)=A(q)/\sqrt{1-q}$ diverges and $\max_s 2s/\cosh s$ governs the radius of convergence in the rescaled expansion parameter $\beta\sqrt{1-q}$.

(2) Zero structure and product representation. Prove that the zeros of $Z(\mathrm{i}t)$ on the positive real $t$-axis form a strictly increasing sequence $t_1(q)<t_2(q)<\cdots$ of simple zeros, that all zeros of the entire function $Z(\beta)$ lie on the imaginary axis, and that $Z(\beta)=\prod_{i\geq 1}\bigl(1+\beta^2/t_i(q)^2\bigr)$ with a convergent product; in particular, the radius of convergence of the high-temperature series equals $t_1(q)=A(q)/\sqrt{1-q}$, so that deliverables (1) and (2) are mutually consistent.

(3) Limit shape. Determine the limit shape of the Dyck paths underlying the moments $\langle 0|T^{2n}|0\rangle$ in the $q$-oscillator case, in the sense of the Avdoshkin–Dymarsky convention cited by Okuyama (2023): paths of $2n$ up/down steps whose weight is the product of the $q$-oscillator Jacobi coefficients $b_h=\sqrt{(1-q^{h})/(1-q)}$ along the path (the Touchard–Riordan formula being an equivalent closed form of the same moments), with heights rescaled as $h/(2n)$ to a profile $f(t)$, $t\in[0,1]$, $f(0)=f(1)=0$, $f\geq 0$, the limit shape being the dominant (saddle-point/typical) profile. The determination must include identifying the regime(s) of $q$ — fixed $q\in[0,1)$ and/or any scaling of $1-q$ with $n$ — in which a nontrivial deterministic shape exists; a rigorous demonstration that the rescaled profile is degenerate at fixed $q$ is an admissible part of the determination. It must also establish the relation, if any, of that shape to the Brownian motion of the boundary particle in the chord/quantum-group construction, a dynamics that emerges in the $q\to 1$ low-energy limit.

The common objective uniting the three deliverables is a proof-level account of how the chord combinatorics of DSSYK determines the singularity structure of $Z(\beta)$: the moments are the Dyck-path sums of (3), the $f_n$ are their connected part in (1), and the zeros control the generating function's singularities in (2). A rigorous refutation of any part — for example, a certified zero of $Z(\beta)$ off the imaginary axis at some $q$, a certified violation of the stated asymptotic form, or a proof that no nontrivial deterministic limit shape exists in the stated sense — is an equally admissible resolution of that part. (Scope note: the finite-$\lambda$ bulk-spacetime interpretation of DSSYK and possible stringy degrees of freedom are out of scope here.)

References

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