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Razvan Teodorescu

Publications and source records attributed to Razvan Teodorescu.

At least 19 recordsLinked to original sources
arXiv2025

The Lagrangian and symplectic structures of the Kuramoto oscillator model

Despite being under intense scrutiny for 50 years, the Kuramoto oscillator model has remained a quintessential representative of non-equilibrium phase transitions. One of the reasons for its enduring relevance is the apparent lack of an optimization formulation, due to the fact that (superficially), the equations of motion seem to not be compatible with a Lagrangian structure. We show that, as a mean-field classical (twisted) spin model on $S^2$, the Kuramoto model can be described variationaly. Based on this result perturbation analysis around (unstable) Kuramoto equilibria are shown to be equivalent to low-energy fluctuations of mean-field Heisenberg spin models. Intriguingly, off-plane perturbations around these equilibria configurations turn out to be described by a semiclassical Gaudin model, pointing to the fact that oscillator synchronization maps to the spin pairing mechanism investigated by Richardson and subsequently by others.

math-ph↗
arXiv2024

Restricted isometric compression of sparse datasets into low-dimensional varieties

This article extends the known restricted isometric projection of sparse datasets in Euclidean spaces $\mathbb{R}^N$ down into low-dimensional subspaces $\mathbb{R}^k, k \ll N,$ to the case of low-dimensional varieties $\mathcal{M} \subset \mathbb{R}^N,$ of codimension $N - k = ω(N)$. Applications to structured/hierarchical datasets are considered.

math-ph↗
arXiv2024

Integrability-preserving regularizations of Laplacian Growth

The Laplacian Growth (LG) model is known as a universality class of scale-free aggregation models in two dimensions, characterized by classical integrability and featuring finite-time boundary singularity formation. A discrete counterpart, Diffusion-Limited Aggregation (or DLA), has a similar local growth law, but significantly different global behavior. For both LG and DLA, a proper description for the scaling properties of long-time solutions is not available yet. In this note, we outline a possible approach towards finding the correct theory yielding a regularized LG and its relation to DLA.

math-ph↗
arXiv2024

Projective connections and extremal domains for analytic content

This note expands on the recent proof \cite{ABKT} that the extremal domains for analytic content in two dimensions can only be disks and annuli. This result's unexpected implication for theoretical physics is that, for extremal domains, the analytic content is a measure of non-commutativity of the (multiplicative) adjoint operators $T, T^†$, where $T^† = \bar z$, and therefore of the quantum deformation parameter (``Planck's constant"). The annular solution (which includes the disk as a special case) is, in fact, a continuous family of solutions, corresponding to all possible positive values of the deformation parameter, consistent with the physical requirement that conformal invariance in two dimensions forbids the existence of a special length scale.

math-ph↗
arXiv2019

Effective distribution of codewords for Low Density Parity Check Cycle codes in the presence of disorder

We review the zeta-function representation of codewords allowed by a parity-check code based on a bipartite graph, and then investigate the effect of disorder on the effective distribution of codewords. The randomness (or disorder) is implemented by sampling the graph from an ensemble of random graphs, and computing the average zeta function of the ensemble. In the limit of arbitrarily large size for the vertex set of the graph, we find an exponential decay of the likelihood for nontrivial codewords corresponding to graph cycles. This result provides a quantitative estimate of the effect of randomization in cybersecurity applications.

math.PR↗
arXiv2015

Efficient algorithms for topological inference on random graphs

In this study, we investigate the problem of classifying, characterizing, and designing efficient algorithms for hard inference problems on planar graphs, in the limit of infinite size. The problem is considered hard if, for a deterministic graph, it belongs to the NP class of computational complexity. A typical example rich in applications is that of connectivity loss in evacuation models for natural hazards management (e.g. coastal floods, hurricanes). Algorithmically, this model reduces to solving a min-cut (or max-flow) problem, with is known to be intractable. The current work covers several generalizations: posing the same problem for non-directed networks subject to random fluctuations (specifically, random graphs from the Erdös-Rényi class); finding efficient convex classifiers for the associated decision problem (deciding whether the graph had become disconnected or not); and the role played by choice of topology (on the space of random graphs) in designing efficient, convex approximation algorithms (in the infinite-size limit of the graph).

math.ST↗
arXiv2015

Braid group representations and cold Fermi gases in the fast pairing regime

It is widely recognized that the main difficulty in designing devices which could process information using quantum states is due to the decoherence of local excitations about a ground state. A solution to this problem was suggested in \cite{Kitaev}, relying on (non-local) topological excitations, structurally protected against local noise. However, a practical implementation of this proposal using special Landau levels in fractional quantum Hall effect systems (FQHE) \cite{QHE} has proven elusive, while accessible FQHE states are theoretically not optimal because their representations in the Hilbert space of states are not dense. We propose using a different physical system (cold Fermi atoms), whose semiclassical dynamics is described by a hyperelliptic function in the Sklyanin formalism. The homological structure of the complex curve corresponds to representations of the braid group, with the action of Hecke operators leading to singularities detectable in the semiclassical oscillations. We argue that, for a fixed genus of the hyperelliptic curve, the Richardson-Gaudin pairing Hamiltonian problem is the singular limit $m \to \infty$ of level-$k$ $\widehat{sl}(2)$, with $k+2 = \frac{4}{8m+1}\to 0$, so that the level $k$ is admissible in the sense of Kac and Kazhdan \cite{KK}, but the corresponding Hecke algebra is a $q-$deformation of the symmetric group with fixed $q = e^{iπ/4}$, as $m \to \infty$.

math-ph↗
arXiv2014

Topological constraints in geometric deformation quantization on domains with multiple boundary components

A topological constraint on the possible values of the universal quantization parameter is revealed in the case of geometric quantization on (boundary) curves diffeomorphic to $S^1$, analytically extended on a bounded domain in $\mathbb{C}$, with $n \ge 2$ boundary components. Unlike the case of one boundary component (such as the canonical Berezin quantization of the Poincaré upper-half plane or the case of conformally-invariant 2D systems), the more general case considered here leads to a strictly positive minimum value for the quantization parameter, which depends on the geometrical data of the domain (specifically, the total area and total perimeter in the smooth case). It is proven that if the lower bound is attained, then $n=2$ and the domain must be annular, with a direct interpretation in terms of the global monodromy.

math-ph↗
arXiv2013

An Overdetermined Problem in Potential Theory

We investigate a problem posed by L. Hauswirth, F. Hélein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.

math.CV↗
arXiv2010

Universal limits of nonlinear measure redistribution processes and their applications

Deriving the time evolution of a distribution of probability (or a probability density matrix) is a problem encountered frequently in a variety of situations: for physical time, it could be a kinetic reaction study, while identifying time with the number of computational steps gives a typical picture of algorithms routinely used in quantum impurity solvers, density functional theory, etc. Using a truncation scheme for the expansion of the exact quantity is necessary due to constraints of the numerical implementation. However, this leads in turn to serious complications such as the Fermion Sign Problem (essentially, density or weights will become negative). By integrating angular degrees of freedom and reducing the dynamics to the radial component, the time evolution is reformulated as a nonlinear integral transform of the distribution function. A canonical decomposition into orthogonal polynomials leads back to the original sign problem, but using a characteristic-function representation allows to extract the asymptotic behavior, and gives an exact large-time limit, for many initial conditions, with guaranteed positivity.

math.PR↗
arXiv2010

Weak solution of the Hele-Shaw problem: shocks and viscous fingering

In Hele-Shaw flows, boundaries between fluids develop unstable viscous fingers. At vanishing surface tension, the fingers further evolve to cusp-like singularities. We show that the problem admits a {\it weak solution} where shock fronts triggered by a singularity propagate together with a fluid. Shocks form a growing, branching tree of a mass deficit, and a line distribution of vorticity where pressure and velocity of the fluid have finite discontinuities. Imposing that the flow remain curl-free at macroscale determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity -- an elementary branching event.

cond-mat.soft↗
arXiv2010

Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers [1, 2] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painleve I equation. We argue that the Painleve I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painleve linear problem.

nlin.SI↗
arXiv2009

Non-Equilibrium Thermodynamics and Topology of Currents

In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cramér functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points.

cond-mat.stat-mech↗
arXiv2008

Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that the ill-defined problem admits a weak {\it dispersive} solution when singularities give rise to a graph of shock waves propagating in the viscous fluid. The graph of shocks grows and branches. Velocity and pressure jump across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2,3) cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.

nlin.SI↗
arXiv2008

Optimal approximation of harmonic growth clusters by orthogonal polynomials

Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoretical model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows, [1]), to the granular dynamics of hard spheres [2], and even diffusion-limited aggregation [3]. Although a complete solution for the continuum case exists [4, 5], efficient approximations of the boundary evolution are very useful due to their practical applications [6]. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel [7] is discussed, as well as relations to potential theory.

math-ph↗
arXiv2008

A linear path toward synchronization: Anomalous scaling in a new class of exactly solvable Kuramoto models

Using a recently introduced linear reformulation of the Kuramoto model of self-synchronizing oscillator systems (arXiv:0704.1166), we study a new class of analytically solvable oscillator systems defined by a particular coupling scheme. We show that these systems have a logarithimic scaling law in the vicinity of the critical point, which may be seen as anomalous with respect to the usual power-law behavior exhibited by the standard Kuramoto model.

cond-mat.stat-mech↗
arXiv2008

Random Matrices in 2D, Laplacian Growth and Operator Theory

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.

nlin.SI↗