arXiv subjects

Oleg Evnin

Publications and source records attributed to Oleg Evnin.

At least 19 recordsLinked to original sources

arXiv2026

Quantum estimates for classical polynomial optimization

The problem of finding lower and upper bounds on multivariate homogeneous polynomials is both difficult and important given its applications to questions ranging from dynamical stability in complex potential landscapes to data analysis. From the standpoint of tensor eigenvalue theory, the question is equivalent to finding the smallest and the largest eigenvalues of the coefficient tensor corresponding to the given polynomial. Standard approaches outlined in the literature amount to running nonlinear iterations in search for the optimal rays along which the growth of the polynomial is fastest or slowest. Unlike the case of matrices (or their corresponding multivariate quadratic forms) convergence of such algorithms for higher-rank tensors is capricious due to the complex topography of polynomial objective functions. In this essay, a very different strategy, inspired by quantum-mechanical variational methods, is introduced for finding bounds on polynomials. The original polynomial is replaced by an operator acting in a suitably chosen (large) space of states, such that in an appropriate "classical" limit this operator approaches the original polynomial expression made of commutative variables. As a result, approximating the smallest and largest eigenvalues of the coefficient tensor, and thus finding bounds on polynomials, amounts to diagonalizing the resulting quantum operator, represented as a large matrix, and then inspecting the smallest and largest eigenvalues of this matrix. This approach is then successfully applied to standard test examples from tensor eigenvalue literature and other problems of interest in mathematical physics including Strichartz-type inequalities.

math-ph↗

arXiv2026

Random matrix theory of integrability-to-chaos transition

The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.

cond-mat.stat-mech↗

arXiv2025

Random matrix theory of sparse neuronal networks with heterogeneous timescales

Training recurrent neuronal networks consisting of excitatory (E) and inhibitory (I) units with additive noise for working memory computation slows and diversifies inhibitory timescales, leading to improved task performance that is attributed to emergent marginally stable equilibria [PNAS 122 (2025) e2316745122]. Yet the link between trained network characteristics and their roles in shaping desirable dynamical landscapes remains unexplored. Here, we investigate the Jacobian matrices describing the dynamics near these equilibria and show that they are sparse, non-Hermitian rectangular-block matrices modified by heterogeneous synaptic decay timescales and activation-function gains. We specify a random matrix ensemble that faithfully captures the spectra of trained Jacobian matrices, arising from the inhibitory core - excitatory periphery network motif (pruned E weights, broadly distributed I weights) observed post-training. An analytic theory of this ensemble is developed using statistical field theory methods: a Hermitized resolvent representation of the spectral density processed with a supersymmetry-based treatment in the style of Fyodorov and Mirlin. In this manner, an analytic description of the spectral edge is obtained, relating statistical parameters of the Jacobians (sparsity, weight variances, E/I ratio, and the distributions of timescales and gains) to near-critical features of the equilibria essential for robust working memory computation.

q-bio.NC↗

arXiv2025

Spontaneous symmetry breaking on graphs and lattices

Spontaneous symmetry breaking is a cornerstone of modern physics, defining a wealth of phenomena in condensed-matter and high-energy physics, and beyond. It requires an infinite number of degrees of freedom, and even then, for continuous symmetries, it only works if the spatial dimension is not too low, following the classic results of Coleman, Hohenberg, Mermin and Wagner. While usually discussed in the context of quantum and statistical field theories, and in particular, effective field theories, there are advantages in addressing the same kind of phenomena on discrete geometric structures rather than conventional manifolds. When the space is discretized into a lattice, a lucid picture of conventional spontaneous symmetry breaking springs up, with the ultraviolet issues of continuum quantum field theory out-of-sight, and the key effect, which is infrared in nature, revealed through elementary harmonic oscillator networks. From there, it is natural to generalize lattices to other graphs/networks. In this setting, the presence of spontaneous symmetry breaking is controlled by fractional generalizations of resistance distance and the Kirchhoff index, and most broadly by the spectral dimension. Predictably, because of the richness of discrete geometric structures in comparison with continuous manifolds, a broader array of geometries emerge where spontaneous breaking of continuous symmetries is blocked by large fluctuations.

cond-mat.dis-nn↗

arXiv2025

Unified theory of local integrals of motion

Conservation laws are of paramount importance in our understanding of classical and quantum dynamics. Here, we present a general framework for constructing exact quantum integrals of motion with the desired locality and quantum numbers, which will be illustrated for the case of many-body-localization (MBL). The latter has been understood theoretically in terms of the existence of an extensive number of local integrals of motion (LIOMs). Using our approach, we show that one can express the task of finding LIOMs as an optimization problem. For some specifications, this problem surprisingly connects to the question of finding classical ground states of spin-glass models. Our work unifies previous results obtained in the MBL context and reveals intriguing connections between many-body localization, spin-glass physics, and constrained optimization problems.

cond-mat.dis-nn↗

arXiv2025

A superintegrable quantum field theory

G\'erard and Grellier proposed, under the name of the cubic Szeg\H{o} equation, a remarkable classical field theory on a circle with a quartic Hamiltonian. The Lax integrability structure that emerges from their definition is so constraining that it allows for writing down an explicit general solution for prescribed initial data, and at the same time, the dynamics is highly nontrivial and involves turbulent energy transfer to arbitrarily short wavelengths. The quantum version of the same Hamiltonian is even more striking: not only the Hamiltonian itself, but also its associated conserved hierarchies display purely integer spectra, indicating a structure beyond ordinary quantum integrability. Here, we initiate a systematic study of this quantum system by presenting a mixture of analytic results and empirical observations on the structure of its eigenvalues and eigenvectors, conservation laws, ladder operators, etc.

nlin.SI↗

arXiv2025

A set of master variables for the two-star random graph

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such `master variables' are usually highly desirable in treatments of `large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.

cond-mat.stat-mech↗

arXiv2025

Analytic and numerical toolkit for the Anderson model in one dimension

The Anderson model in one dimension is a quantum particle on a discrete chain of sites with nearest-neighbor hopping and random on-site potentials. It is a progenitor of many further models of disordered systems, and it has spurred numerous developments in various branches of physics. The literature does not readily provide, however, practical analytic tools for computing the density-of-states of this model when the distribution of the on-site potentials is arbitrary. Here, supersymmetry-based techniques are employed to give an explicit linear integral equation whose solutions control the density-of-states. The output of this analytic procedure is in perfect agreement with numerical sampling. By Thouless formula, these results immediately provide analytic control over the localization length.

cond-mat.dis-nn↗

arXiv2025

Statistical field theory of random graphs with prescribed degrees

Statistical field theory methods have been very successful with a number of random graph and random matrix problems, but it is challenging to apply these methods to graphs with prescribed degree sequences due to the extensive number of constraints that enforce the desired degree at each vertex. Building on top of recent results where similar methods are applied to random regular graph counting, we develop an accurate statistical field theory description for the adjacency matrix spectra of graphs with prescribed degrees. For large graphs, the expectation values are dominated by functional saddle points satisfying explicit equations. For the case of regular graphs, this immediately leads to the known McKay distribution. We then consider mixed-regular graphs with N1 vertices of degree d1, N2 vertices of degree d2, etc, such that the ratios N_i/N are kept fixed as N goes to infinity. For such graphs, the eigenvalue densities are governed by nonlinear integral equations of the Hammerstein type. Solving these equations numerically reproduces with an excellent accuracy the empirical eigenvalue distributions.

cond-mat.stat-mech↗

arXiv2025

Ensemble inequivalence and phase transitions in unlabeled networks

We discover a first-order phase transition in the canonical ensemble of random unlabeled networks with a prescribed average number of links. The transition is caused by the nonconcavity of microcanonical entropy. Above the critical point coinciding with the graph symmetry phase transition, the canonical and microcanonical ensembles are equivalent and have a well-behaved thermodynamic limit. Below the critical point, the ensemble equivalence is broken, and the canonical ensemble is a mixture of phases: empty networks and networks with average degrees diverging logarithmically with the network size. As a consequence, networks with bounded average degrees do not survive in the thermodynamic limit, decaying into the empty phase. The celebrated percolation transition in labeled networks is thus absent in unlabeled networks. In view of these differences between labeled and unlabeled ensembles, the question of which one should be used as a null model of different real-world networks cannot be ignored.

cond-mat.stat-mech↗

arXiv2025

First return times on sparse random graphs

We consider random walks in the form of nearest-neighbor hopping on Erdos-Renyi random graphs of finite fixed mean degree c as the number of vertices N tends to infinity. In this regime, using statistical field theory methods, we develop an analytic theory of the first return time probability distribution. The problem turns out closely related to finding the spectrum of the normalized graph Laplacian that controls the continuum time version of the nearest-neighbor-hopping random walk. In the infinite graph limit, where loops are highly improbable, the returns operate in a manner qualitatively similar to c-regular trees, and the expressions for probabilities resemble those on random c-regular graphs. Because the vertex degrees are not exactly constant, however, the way c enters the formulas differs from the dependence on the graph degree of first return probabilities on random regular graphs.

cond-mat.dis-nn↗

arXiv2025

Multiseed Krylov complexity

Krylov complexity is an attractive measure for the rate at which quantum operators spread in the space of all possible operators under dynamical evolution. One expects that its late-time plateau would distinguish between integrable and chaotic dynamics, but its ability to do so depends precariously on the choice of the initial seed. We propose to apply such considerations not to a single operator, but simultaneously to a collection of initial seeds in the manner of the block-Lanczos algorithm. We furthermore suggest that this collection should comprise all simple (few-body) operators in the theory, which echoes the applications of Nielsen complexity to dynamical evolution. The resulting construction, unlike the conventional Krylov complexity, reliably distinguishes integrable and chaotic Hamiltonians without any need for fine-tuning.

quant-ph↗

arXiv2025

Hammerstein equations for sparse random matrices

Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.

cond-mat.dis-nn↗

arXiv2025

Phase-space localization at the lowest Landau level

We consider bosons with weak contact interactions in a harmonic trap and focus on states at the lowest Landau level. Motivated by the known nontrivial phase-space topography of the energy functional of the corresponding Gross-Pitaevskii equation, we explore Husimi distributions of quantum energy eigenstates in the classical phase space of the Schroedinger field. With interactions turned off, the energy levels are highly degenerate and the Husimi distributions do not manifest any particular localization properties. With interactions turned on, the degeneracy is lifted, and a selection of energy levels emerges whose Husimi distributions are localized around low-dimensional surfaces in the phase space.

cond-mat.quant-gas↗

arXiv2024

A Gaussian integral that counts regular graphs

In a recent article J. Phys. Compl. 4 (2023) 035005, Kawamoto evoked statistical physics methods for the problem of counting graphs with a prescribed degree sequence. This treatment involved truncating a particular Taylor expansion at the first two terms, which resulted in the Bender-Canfield estimate for the graph counts. This is surprisingly successful since the Bender-Canfield formula is asymptotically accurate for large graphs, while the series truncation does not a priori suggest a similar level of accuracy. We upgrade the above treatment in three directions. First, we derive an exact formula for counting d-regular graphs in terms of a d-dimensional Gaussian integral. Second, we show how to convert this formula into an integral representation for the generating function of d-regular graph counts. Third, we perform explicit saddle point analysis for large graph sizes and identify the saddle point configurations responsible for graph count estimates. In these saddle point configurations, only two of the integration variables condense to significant values, while the remaining ones approach zero for large graphs. This provides an underlying picture that justifies Kawamoto's earlier findings.

cond-mat.stat-mech↗

arXiv2024

A relation between Krylov and Nielsen complexity

Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.

quant-ph↗

arXiv2024

Integrability and complexity in quantum spin chains

There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.

quant-ph↗

arXiv2023

Democracy from topology

Chiral form fields in $d$ dimensions can be effectively described as edge modes of topological Chern-Simons theories in $d+1$ dimensions. At the same time, manifestly Lorentz-invariant Lagrangian description of such fields directly in terms of a $d$-dimensional field theory is challenging and requires introducing nontrivial auxiliary gauge fields eliminated on-shell with extra gauge symmetries. A recent work by Arvanitakis et al.\ demonstrates (emphasizing the case of 2d chiral bosons) that the two approaches are related, and a peculiar reduction on the $(d+1)$-dimensional topological Lagrangian automatically leads to $d$-dimensional Lagrangians with appropriate sets of auxiliary fields. We develop this setup in three distinct directions. First, we demonstrate how arbitrary Abelian self-interactions for chiral forms can be included using nonlinear boundary terms in the Chern-Simons theory. Second, by generalizing the Chern-Simons theory to the BF theory, we obtain an analogous democratic description of non-chiral form fields, where electric and magnetic potentials appear as explicit dynamical variables. Third, we discuss the effects of introducing topological interactions in the higher-dimensional bulk, which produce extra interaction terms in the boundary theory. When applied to a topological 4-form field in 12 dimensions, this construction results in a democratic description of the 3-form gauge field of the 11-dimensional supergravity.

hep-th↗