We establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-$N$ and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent, with the precise mapping $b(1-x)=\sqrt{R(x)}$ and $a(1-x)=S(x)$. 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 establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-N and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent, with the precise mapping b(1−x)=R(x) and 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 study the spread complexity of the infinite-temperature thermofield double state in large-N random matrix theory and investigate the physical meaning of its Lanczos coefficients bn. The leading density of states defines a family of orthogonal polynomials whose recursion coefficients coincide with the Lanczos coefficients of the associated Krylov dynamics. Starting from the Coulomb-gas equation, we derive a pair of nonlinear relations for these coefficients, which we call the Lanczos equations. For normalizable one-cut ensembles, the Lanczos coefficients approach constants at large Krylov index, rendering the asymptotic Krylov chain translationally invariant and the late-time growth of spread complexity linear, with the growth rate given by the spectral average of the group velocity obtained from a Bloch-wave analysis. For an even potential, we further show that the difference between consecutive squared Lanczos coefficients, bn+12−bn2, is the generating function for two-legged maps whose marked legs are separated by the exact geodesic distance n. Consequently, the spread complexity acceleration operator equals twice the geodesic generating operator. We illustrate these results in the quartic matrix model, where the expansion in the quartic coupling counts tetravalent planar maps, and in the double-scaled Sachdev--Ye--Kitaev model, where the auxiliary Hilbert space can be identified with the physical chord Hilbert space.
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 Nwd=1, the leading asymptotic form of Rn 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 Rn and Sn, whereas the DSSYK model can exhibit multiple transition regions in Rn, 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.
The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of Random Matrix Theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulae relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in Thermo-Field Double states and the spectral form factor for Gaussian and Non-Gaussian RMTs.
This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around t=0. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a ``proof of principle'' for using Krylov complexity as a complete characterization of operator evolution in quantum systems.
The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulas relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.
A bstract Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ( C K ) and Spread complexity ( C S ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators . After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between C K and 2 C S . Furthermore, for maximally entangled pure states, we find that the moment-generating function of C K becomes the Spectral Form Factor and, at late-times, C K is simply related to NC S for N ≥ 2 within the N -dimensional Hilbert space. Notably, we confirm that C K = 2 C S holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.