arXiv subjects

Joseph W. Baron

Publications and source records attributed to Joseph W. Baron.

At least 19 recordsLinked to original sources

arXiv2026

Lecture notes on random matrix theory: the results, the applications, and the analytical tools

Random matrix theory has established itself as a theoretical cornerstone of the mathematical sciences over the past century. It has undeniable utility in areas of research as diverse as nuclear physics, finance, ecology and disordered systems. The purpose of these notes is twofold. First, the most famous and widely used classic results are derived in a pedagogical manner, mostly using the comparatively elementary and transparent cavity method. The significance of each result is then demonstrated in the context of a particular application. There are also some select exercises at the end of each section. In the second part of these notes, a reference guide of analytical techniques for the random-matrix/disordered-systems practitioner is provided. Introducing the diagrammatic, replica, path-integral, and supersymmetric formalisms from first principles, we rederive some of the aforementioned classic results, particularly focussing on the simplest one -- the semicircle law. Innovations such as the population dynamics method and the tools of free probability theory are also included. We discuss the merits of each analytical approach, and we highlight the contexts in which each becomes particularly useful.

cond-mat.dis-nn↗

arXiv2026

Extinction drives emergent metastability in complex ecosystems

Extinction is inevitable; every species eventually dies out, impacting the ecosystem it is part of. Over the past few decades, extensive research stemming from the stability-diversity debate has addressed how species diversity contributes to the stability of large ecosystems. However, conventional stability criteria often rely on deterministic frameworks, overlooking the intrinsic population fluctuations that allow any species to go extinct by chance. In this paper, we incorporate demographic stochasticity into large complex ecosystems with a rule-based model. We demonstrate that such demographic fluctuations rapidly prune the low-abundance species from the ecological community, thereby securing higher systemic stability. By developing a bottom-up theory to characterise the statistics of extinction dynamics, we discover that the fraction of surviving species exhibits an anomalous heavy-tailed decay over time, revealing the emergence of remnant communities with robust metastability. Our results highlight that when demographic fluctuations are accounted for, ecosystems self-stabilise by reducing their diversity even when they are predicted to be chaotic in the deterministic limit.

q-bio.PE↗

arXiv2026

Local density of states distribution and multifractal eigenvectors of weighted random networks via the cavity approach

We study the local density of states (LDoS) distribution of a general class of weighted Erdős-Rényi graphs. Using the cavity method, we obtain a good approximation to the full LDoS distribution and compact expressions for its power-law tails, which we show to have exponent $3$ in the extended phase. We deduce that the eigenvectors in the continuous part of the spectrum are extended but (weakly) multifractal, and we extract expressions for the associated fractal dimensions and the singularity spectrum. We also demonstrate that the inverse participation ratio in this multifractal phase exhibits an unusual logarithmic scaling with system size, which is neither fully-extended nor localised by the usual definitions. Finally, we verify that some symmetry properties (derived from the non-linear sigma model), which have been shown to hold for many systems exhibiting multifractality, also hold in our case, both for the LDoS distribution and the singularity spectrum.

cond-mat.dis-nn↗

arXiv2025

Eigenvalue spectral tails and localization properties of asymmetric networks

In contrast to the neatly bounded spectra of densely populated large random matrices, sparse random matrices often exhibit unbounded eigenvalue tails on the real and imaginary axis, called Lifshitz tails. In the case of asymmetric matrices, concise mathematical results have proved elusive. In this work, we present an analytical approach to characterising these tails. We exploit the fact that eigenvalues in the tail region have corresponding eigenvectors that are exponentially localised on highly-connected hubs of the network associated to the random matrix. We approximate these eigenvectors using a series expansion in the inverse connectivity of the hub, where successive terms in the series take into account further sets of next-nearest neighbours. By considering the ensemble of such hubs, we are able to characterise the eigenvalue density and the extent of localisation in the tails of the spectrum in a general fashion. As such, we classify a number of different asymptotic behaviours in the Lifshitz tails, as well as the leading eigenvalue and the inverse participation ratio. We demonstrate how an interplay between matrix asymmetry, network structure, and the edge-weight distribution leads to the variety of observed behaviours.

cond-mat.dis-nn↗

arXiv2025

Classes of non-Gaussian random matrices: long-range eigenvalue correlations and non-ergodic extended eigenvectors

The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown to be robust to many changes in the random matrix ensemble, and are often well-predicted by results corresponding to Gaussian random matrices in many applications. In this work, we show that, in contrast, the long-range correlations of the eigenvalues of random matrices are more sensitive. Using a path-integral approach, we identify classes of statistical deviations from the Gaussian Orthogonal random matrix Ensemble (GOE) that give rise to long-range correlations. We provide closed-form analytical expressions for the eigenvalue compressibility and two-point correlations, which ordinarily vanish for the GOE, but are non-zero here. These expressions are universal in the limit of small non-Gaussianity. We discuss how these results suggest the presence of non-ergodic eigenvectors, and we verify numerically that the eigenvector component distributions of a wide variety of non-Gaussian ensembles exhibit the associated power-law tails. We also comment on how these findings reveal the need to go beyond simple mean-field theories in disordered systems with non-Gaussian interactions.

cond-mat.dis-nn↗

arXiv2025

A path integral approach to sparse non-Hermitian random matrices

The theory of large random matrices has proved an invaluable tool for the study of systems with disordered interactions in many quite disparate research areas. Widely applicable results, such as the celebrated elliptic law for dense random matrices, allow one to deduce the statistical properties of the interactions in a complex dynamical system that permit stability. However, such simple and universal results have so far proved difficult to come by in the case of sparse random matrices. Here, we perform an expansion in the inverse connectivity, and thus derive general modified versions of the classic elliptic and semi-circle laws, taking into account the sparse correction. This is accomplished using a dynamical approach, which maps the hermitized resolvent of a random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling one to construct Feynman diagrams, which facilitate the perturbative analysis. Additionally, in order to demonstrate the broad utility of the path integral framework, we derive a generic non-Hermitian generalization of the Marchenko-Pastur law, and we also show how one can handle non-negligible higher-order statistics (i.e. non-Gaussian statistics) in dense ensembles.

cond-mat.dis-nn↗

arXiv2025

Breakdown of random-matrix universality in persistent Lotka--Volterra communities

The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this letter, we offer a pertinent counterexample in the context of the generalised Lotka--Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of `feasible' ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.

cond-mat.dis-nn↗

arXiv2025

Dispersal-induced instability in complex ecosystems

In his seminal work in the 1970s, Robert May suggested that there is an upper limit to the number of species that can be sustained in stable equilibrium by an ecosystem. This deduction was at odds with both intuition and the observed complexity of many natural ecosystems. The so-called stability-diversity debate ensued, and the discussion about the factors contributing to ecosystem stability or instability continues to this day. We show in this work that dispersal can be a destabilising influence. To do this, we combine ideas from Alan Turing's work on pattern formation with May's random-matrix approach. We demonstrate how a stable equilibrium in a complex ecosystem with trophic structure can become unstable with the introduction of dispersal in space, and we discuss the factors which contribute to this effect. Our work highlights that adding more details to the model of May can give rise to more ways for an ecosystem to become unstable. Making May's simple model more realistic is therefore unlikely to entirely remove the upper bound on complexity.

q-bio.PE↗

arXiv2025

Eigenvalues of random matrices with generalised correlations: a path integral approach

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalised correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.

cond-mat.dis-nn↗

arXiv2024

Interaction networks in persistent Lotka-Volterra communities

A central concern of community ecology is the interdependence between interaction strengths and the underlying structure of the network upon which species interact. In this work we present a solvable example of such a feedback mechanism in a generalised Lotka-Volterra dynamical system. Beginning with a community of species interacting on a network with arbitrary degree distribution, we provide an analytical framework from which properties of the eventual `surviving community' can be derived. We find that highly-connected species are less likely to survive than their poorly connected counterparts, which skews the eventual degree distribution towards a preponderance of species with low degree, a pattern commonly observed in real ecosystems. Further, the average abundance of the neighbours of a species in the surviving community is lower than the community average (reminiscent of the famed friendship paradox). Finally, we show that correlations emerge between the connectivity of a species and its interactions with its neighbours. More precisely, we find that highly-connected species tend to benefit from their neighbours more than their neighbours benefit from them. These correlations are not present in the initial pool of species and are a result of the dynamics.

q-bio.PE↗

arXiv2023

Mean-Field Analysis of the Glassy Dynamics of an Elastoplastic Model of Super-Cooled Liquids

We present a mean-field theory of a coarse-grained model of a super-cooled liquid in which relaxation occurs via local plastic rearrangements. Local relaxation can be induced by thermal fluctuations or by the long-range elastic consequences of other rearrangements. We extract the temperature dependence of both the relaxation time and the lengthscale of dynamical correlations. We find two dynamical regimes. First, a regime in which the characteristic time and length scales diverge as a power law at a critical temperature $T_c$. This regime is found by an approximation that neglects activated relaxation channels, which can be interpreted as akin to the one found by the mode-coupling transition of glasses. In reality, only a cross-over takes place at $T_c$. The residual plastic activity leads to a second regime characterised by an Arrhenius law below $T_c$. In this case, we show that the lengthscale governing dynamical correlations diverges as a power law as $T\to 0$, and is logarithmically related to the relaxation time.

cond-mat.soft↗

arXiv2023

Eigenvalue spectra of finely structured random matrices

Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are applicable in a myriad of different contexts. However, it is often assumed that all components of the complex system in question are statistically equivalent, which is unrealistic in many applications. Here, we introduce the concept of a finely-structured random matrix. These are random matrices with element-specific statistics, which can be used to model systems in which the individual components are statistically distinct. By supposing that the degree of `fine structure' in the matrix is small, we arrive at a succinct `modified' elliptical law. We demonstrate the direct applicability of our results to the niche and cascade models in theoretical ecology, as well as a model of a neural network, and a directed network with arbitrary degree distribution. The simple closed form of our central results allow us to draw broad qualitative conclusions about the effect of fine structure on stability.

cond-mat.dis-nn↗

arXiv2022

Analytical and numerical treatment of continuous ageing in the voter model

The conventional voter model is modified so that an agent's switching rate depends on the `age' of the agent, that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. Lewis' thinning algorithm can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

physics.soc-ph↗

arXiv2022

Generalised Lotka-Volterra model with hierarchical interactions

In the analysis of complex ecosystems it is common to use random interaction coefficients, often assumed to be such that all species are statistically equivalent. In this work we relax this assumption by choosing interactions according to the cascade model, which we incorporate into a generalised Lotka-Volterra dynamical system. These interactions impose a hierarchy in the community. Species benefit more, on average, from interactions with species further below them in the hierarchy than from interactions with those above. Using dynamic mean-field theory, we demonstrate that a strong hierarchical structure is stabilising, but that it reduces the number of species in the surviving community, as well as their abundances. Additionally, we show that increased heterogeneity in the variances of the interaction coefficients across positions in the hierarchy is destabilising. We also comment on the structure of the surviving community and demonstrate that the abundance and probability of survival of a species is dependent on its position in the hierarchy.

q-bio.PE↗

arXiv2022

Eigenvalue spectra and stability of directed complex networks

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build upon previous results, which usually only take into account the mean degree of the network, by allowing for non-trivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulae for the corrections (due to non-zero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semi-circle law, the Girko circle law and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of an directed Barabasi-Albert network. We are thus able to make general deductions about the effect of network heterogeneity on the stability of complex dynamic systems.

cond-mat.dis-nn↗

arXiv2021

Sampling rare trajectories using stochastic bridges

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rate trajectories. Further, we also compare the stochastic bridges produced using our method to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the paths produced using our method, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that our method can be used to judge the accuracy of the WKB approximation at finite levels of noise.

cond-mat.stat-mech↗

arXiv2021

Persistent individual bias in a voter model with quenched disorder

Many theoretical studies of the voter model (or variations thereupon) involve order parameters that are population-averaged. While enlightening, such quantities may obscure important statistical features that are only apparent on the level of the individual. In this work, we ask which factors contribute to a single voter maintaining a long-term statistical bias for one opinion over the other in the face of social influence. To this end, a modified version of the network voter model is proposed, which also incorporates quenched disorder in the interaction strengths between individuals and the possibility of antagonistic relationships. We find that a sparse interaction network and heterogeneity in interaction strengths give rise to the possibility of arbitrarily long-lived individual biases, even when there is no population-averaged bias for one opinion over the other. This is demonstrated by calculating the eigenvalue spectrum of the weighted network Laplacian using the theory of sparse random matrices.

physics.soc-ph↗

arXiv2020

Effective diffusion coefficients in reaction-diffusion systems with anomalous transport

We show that the Turing patterns in reaction systems with subdiffusion can be replicated in an effective system with Markovian cross-diffusion. The effective system has the same Turing instability as the original system, and the same patterns. If particles are short-lived, the transient dynamics are captured as well. We use the cross-diffusive system to define effective diffusion coefficients for the system with anomalous transport, and we show how they can be used to efficiently describe the Turing instability. We also demonstrate that the mean squared displacement of a suitably defined ensemble of subdiffusing particles grows linearly with time, with a diffusion coefficient which agrees with our earlier calculations. We verify these deductions by numerically integrating both the fractional reaction-diffusion equation and its normally diffusing counterpart. Our findings suggest that cross-diffusive behaviour can come about as a result of anomalous transport.

nlin.PS↗