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arXiv:2608.10072

Recursion Coefficients and Krylov Dynamics in Polynomial Random Matrix Models

Juan F. Pedraza, Le-Chen Qu

hep-thmath-phquant-phSubmitted Aug 10, 2026• Announced Aug 12, 2026
v1 · Aug 10, 2026v2 · Aug 24, 2026
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Abstract

We study the recursion coefficients of orthogonal polynomials and their associated Krylov dynamics in random matrix models with high-degree and possibly asymmetric polynomial potentials. We develop a moment recursion method that, when combined with the recursive algorithm, provides an efficient construction of the recursion coefficients. We also obtain their large-$n$ asymptotic behavior for general asymmetric potentials; for $Nw_d=1$, the leading asymptotic form of $R_n$ reproduces Freud's conjecture. We apply this framework to an asymmetric quartic potential and to the double-scaled Sachdev-Ye-Kitaev (DSSYK) model. In both models, the recursion functions capture the overall qualitative behavior of the recursion coefficients, and the gradient catastrophes of the recursion functions are associated with ``chaotic'' transition regions in the recursion coefficients. For the quartic potential, such regions can occur in both $R_n$ and $S_n$, whereas the DSSYK model can exhibit multiple transition regions in $R_n$, with the recursion function remaining accurate in the smooth intervals between them. Finally, we compute the corresponding spread complexity and find that transition regions do not qualitatively modify its behavior, while a two branch structure produces early time oscillations followed by monotonic growth.

Comments
16 pages, 7 figures

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We study the recursion coefficients of orthogonal polynomials and their associated Krylov dynamics in random matrix models with high-degree and possibly asymmetric polynomial potentials. We develop a moment recursion method that, when combined with the recursive algorithm, provides an efficient construction of the recursion coefficients. We also obtain their large-nnn asymptotic behavior for general asymmetric potentials; for Nwd=1Nw_d=1Nwd​=1, the leading asymptotic form of RnR_nRn​ reproduces Freud's conjecture. We apply this framework to an asymmetric quartic potential and to the double-scaled Sachdev-Ye-Kitaev (DSSYK) model. In both models, the recursion functions capture the overall qualitative behavior of the recursion coefficients, and the gradient catastrophes of the recursion functions are associated with ``chaotic'' transition regions in the recursion coefficients. For the quartic potential, such regions can occur in both RnR_nRn​ and SnS_nSn​, whereas the DSSYK model can exhibit multiple transition regions in RnR_nRn​, with the recursion function remaining accurate in the smooth intervals between them. Finally, we compute the corresponding spread complexity and find that transition regions do not qualitatively modify its behavior, while a two branch structure produces early time oscillations followed by monotonic growth.
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We establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-NNN and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent, with the precise mapping b(1−x)=R(x)b(1-x)=\sqrt{R(x)}b(1−x)=R(x)​ and a(1−x)=S(x)a(1-x)=S(x)a(1−x)=S(x). As a result, the two formalisms yield identical expressions for the leading density of states. We further analyze the Krylov dynamics associated with the recursion coefficients and show that the orthogonal polynomials admit a natural interpretation as Krylov polynomials. This picture is realized explicitly in the Gaussian Unitary Ensemble, where all quantities can be computed analytically.
We use Krylov complexity as a typical diagnostic of quantum many body dynamics in the context of BMN matrix model at large mass gap. We calculate several physical entities, for example, the moments and return amplitudes at large mass deformation. We calculate them for both spread complexity of states as well as Krylov growth of operators in the matrix model. We propose a general expression for the moments at any given order nnn. We also discuss orthogonal polynomials for spread as well as operator Krylov complexity. In the context of the Krylov operator growth we further compute the spectral function and the density of states. A careful analysis of the spectral function near the resonance reveals various IR divergences in addition to the physical collective modes. These collective modes appear to be exceptionally stable due to large mass gap. On the other hand, the UV of the theory appears to be extremely damped due to higher scattering rates. We carefully diagnose these IR divergences near resonance which reveals that these collective modes are in fact non-propagating and should be thought of as diffusion modes or relaxation modes with infinite relaxation time. We also calculate the Krylov variance and Krylov entropy for the matrix model and in particular at early time. The linear early time growth of the Krylov entropy confirms the onset of quantum chaos in the matrix model at large mass gap.
A bstract We use Krylov complexity to study operator growth in the q -body dissipative Sachdev-Ye-Kitaev (SYK) model, where the dissipation is modeled by linear and random p -body Lindblad operators. In the large q limit, we analytically establish the linear growth of two sets of coefficients for any generic jump operators. We numerically verify this by implementing the bi-Lanczos algorithm, which transforms the Lindbladian into a pure tridiagonal form. We find that the Krylov complexity saturates inversely with the dissipation strength, while the dissipative timescale grows logarithmically. This is akin to the behavior of other 𝔮-complexity measures, namely out-of-time-order correlator (OTOC) and operator size, which we also demonstrate. We connect these observations to continuous quantum measurement processes. We further investigate the pole structure of a generic auto-correlation and the high-frequency behavior of the spectral function in the presence of dissipation, thereby revealing a general principle for operator growth in dissipative quantum chaotic systems.
The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples NNN complex fermions to MMM bosons of frequency ω0ω_0ω0​ through random Yukawa couplings of strength ggg, and its dynamics depends only on R≡ω0/g2/3R\equivω_0/g^{2/3}R≡ω0​/g2/3. For R≪1R\ll1R≪1 the bosons are slow and act as a static random background in which the fermions are free, while for R≫1R\gg1R≫1 they are only virtually excited and generate a random four-fermion interaction of rank MMM, which is generically chaotic for M≥2M\geq2M≥2. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around R≈0.3R\approx0.3R≈0.3, and adding boson modes moves it to smaller RRR. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20%\%% below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger RRR the boson-number bands slow down the spreading of the state in Krylov space.
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