arXiv subjects

Pragya Shukla

Publications and source records attributed to Pragya Shukla.

At least 19 recordsLinked to original sources

arXiv2026

On the synaptic matrix eigenvalues of sparsely connected neural networks

The spectral behaviour of the synaptic matrix, representing the neuronal connection strengths, is an important tool to analyze the stability and transient dynamics of a typical brain as well as its learning process and memory capacity. The complexity of the brain due to large number of neurons as well as underlying transient mechanisms e.g. homeostasis, seizure or synaptic plasticity can lead to networks with time-varying degree and type of sparsity. This renders an exact determination of the synaptic matrix not only technically difficult but also meaningless, leaving its statistical analysis as the best available theoretical approach. This motivates us to pursue a spectral analysis of the synaptic matrix models with different type of sparsity and thereby analyze latter's role on various aspects of network dynamics and stability. Our results have potential relevance for detemining the type of synaptic sparsity required to induce a specific brain function or desired transient mechanism e.g for pharmacological effects or physiological modulators.

q-bio.NC↗

arXiv2026

Entanglement dynamics of many-body quantum states: sensitivity to system conditions and a hidden universality

We consider physical Hamiltonians that can be represented by the multiparametric Gaussian ensembles, theoretically derive the state ensembles for its eigenstates and analyze the effect of varying system conditions on its bipartite entanglement entropy. Our approach leads to a single parametric based common mathematical formulation for the evolution of the entanglement statistics of different states of a given Hamiltonian or different Hamiltonians subjected to same symmetry constraints. The parameter turns out to be a single functional of the system parameters and thereby reveals a deep web of connection hidden underneath different quantum states.

quant-ph↗

arXiv2025

Single-particle entanglement dynamics in complex systems

We analyze the effect of varying system conditions on the single-particle entanglement entropy for an arbitrary eigenstate of a complex system that can be described by a multiparametric Gaussian ensemble. Our theoretical analysis leads to the identification of a single functional of the system parameters that governs the entropy dynamics. This reveals a sensitivity of the entropy to collective information content, characterized by the functional, instead of the individual system details. The functional can further be used to identify the universality classes as well as a deep web of connection underlying different quantum states.

quant-ph↗

arXiv2025

Spectral distribution of sparse Gaussian Ensembles of Real Asymmetric Matrices

Theoretical analysis of biological and artificial neural networks e.g. modelling of synaptic or weight matrices necessitate consideration of the generic real-asymmetric matrix ensembles, those with varying order of matrix elements e.g. a sparse structure or a banded structure. We pursue the complexity parameter approach to analyze the spectral statistics of the multiparametric Gaussian ensembles of real asymmetric matrices and derive the ensemble averaged spectral densities for real as well as complex eigenvalues. Considerations of the matrix elements with arbitrary choice of mean and variances render us the freedom to model the desired sparsity in the ensemble. Our formulation provides a common mathematical formulation of the spectral statistics for a wide range of sparse real-asymmetric ensembles and also reveals, thereby, a deep rooted universality among them.

cond-mat.dis-nn↗

arXiv2024

Average Spectral Density of Multiparametric Gaussian Ensembles of Complex Matrices

A statistical description of part of a many body system often requires a non-Hermitian random matrix ensemble with nature and strength of randomness sensitive to underlying system conditions. For the ensemble to be a good description of the system, the ensemble parameters must be determined from the system parameters. This in turn makes its necessary to analyze a wide range of multi-parametric ensembles with different kinds of matrix elements distributions. The spectral statistics of such ensembles is not only system-dependent but also non-ergodic as well as non-stationary. A change in system conditions can cause a change in the ensemble parameters resulting an evolution of the ensemble density and it is not sufficient to know the statistics for a given set of system conditions. This motivates us to theoretically analyze a multiparametric evolution of the ensemble averaged spectral density of a multiparametric Gaussian ensemble on the complex plane. Our analysis reveals the existence of an evolutionary route common to the ensembles belonging to same global constraint class and thereby derives a complexity parameter dependent formulation of the spectral density for the non-equilibrium regime of the spectral statistics, away from Ginibre equilibrium limit.

cond-mat.dis-nn↗

arXiv2024

Edge of entanglement in non-ergodic states: a complexity parameter formulation

We analyze the subsystem size scaling of the entanglement entropy of a non-ergodic pure state that can be described by a multi-parametric Gaussian ensemble of complex matrices in a bipartite basis. Our analysis indicates, for a given set of global constraints, the existence of infinite number of universality classes of local complexity, characterized by the complexity parameter, for which the entanglement entropy reveals a universal scaling with subsystem size. A rescaling of the complexity parameter helps us to identify the critical regime for the entanglement entropy of a broad range of pure non-ergodic states.

quant-ph↗

arXiv2024

Distribution of the entanglement entropy of a non-ergodic quantum state

We theoretically derive the probability densities of the entanglement measures of a pure non-ergodic many-body state, represented in a bipartite product basis and with its reduced density matrix described by a generalized, multi-parametric Wishart ensemble with unit trace. Our results indicate significant fluctuations of the measures around their average behavior (specifically for the states away from separability and maximum entanglement limits). The information is relevant not only for hierarchical arrangement of entangled states (e.g., revealing the flaws in their characterization based on average behavior) but also for phase transition studies of many body systems.

quant-ph↗

arXiv2024

Non-Hermiticity and Universality

. We study the statistical properties of the eigenvalues of non-Hermitian operators assoicated with the dissipative complex systems. By considering the Gaussian ensembles of such operators, a hierarchical relation between the correlators is obtained. Further the eigenvalues are found to behave like particles moving on a complex plane under 2-body (inverse square) and 3-body interactions and there seems to underlie a deep connection and universality in the spectral behaviour of different complex systems. .

cond-mat.stat-mech↗

arXiv2024

Spectral fluctuations of multiparametric complex matrix ensembles: evidence of a single parameter dependence

We numerically analyze the spectral statistics of the multiparametric Gaussian ensembles of complex matrices with zero mean and variances with different decay routes away from the diagonals. As the latter mimics different degree of effective sparsity among the matrix elements, such ensembles can serve as good models for a wide range of phase transitions e.g. localization to delocalization in non-Hermitian systems or Hermitian to non-Hermitian one. Our analysis reveals a rich behavior hidden beneath the spectral statistics e.g. a crossover of the spectral statistics from Poisson to Ginibre universality class with changing variances for finite matrix size, an abrupt transition for infinite matrix size and the role of complexity parameter, a single functional of all system parameters, as a criteria to determine critical point. We also confirm the theoretical predictions in \cite{psgs, psnh}, regarding the universality of the spectral statistics in non-equilibrium regime of non-Hermitian systems characterized by the complexity parameter.

cond-mat.dis-nn↗

arXiv2022

Entanglement dynamics of multi-parametric random states: a single parametric formulation

A non-ergodic quantum state of a many body system is in general random as well as multi-parametric, former due to a lack of exact information due to complexity and latter reflecting its varied behavior in different parts of the Hilbert space. An appropriate representation for the reduced density matrix of such a state is a generalized, multi-parametric Wishart ensemble with unit trace. Our theoretical analysis of these ensembles not only resolves the controversy about the growth rates of the average information entropies of the generic states but also leads to new insights in their entanglement dynamics. While the state itself is multi-parametric, we find that the growth of the average measures can be described in terms of an information-theoretic function, referred as the complexity parameter. The latter in turn leads to a common mathematical formulation of the measures for a wide range of states; it could also act as a possible tool for hierarchical arrangement of the entangled states of different systems.

quant-ph↗

arXiv2022

Many body density of states in the edge of the spectrum: non-interacting limit

In noninteracting limit, the density of states of a many body system can be expressed as the convolution of single body density of states of its subunits. Here we use the formulation to derive the ensemble averaged many body density of states for the cases in which subunits can be modelled by Gaussian or Wishart random matrix ensembles.

cond-mat.stat-mech↗

arXiv2021

Boson peak in amorphous systems: role of phonon mediated coupling of nano-clusters

Based on a description of an amorphous solid as a collection of coupled nanosize molecular clusters referred as basic blocks, we analyse the statistical properties of its Hamiltonian. The information is then used to derive the ensemble averaged density of the vibrational states (non-phonon) which turns out to be a Gaussian in the bulk of the spectrum and an Airy function in the low frequency regime. A comparison with experimental data for five glasses confirms validity of our theoretical predictions.

cond-mat.stat-mech↗

arXiv2021

Low temperature heat capacity of amorphous systems: physics at nano-scales

Contrary to previous studies of boson peak, we analyze the density of states and specific heat contribution of dispersion forces in an amorphous solid of nano-scales ($\sim 3 nm$). Our analysis indicates a universal semi-circle form of the average density of states in the bulk of the spectrum along with a super-exponentially increasing behavior in its edge. The latter in turn leads to a specific heat, behaving linearly below $T < 1^o \; {\bf K}$ even at nano-scales, and, surprisingly agreeing with the experiments although the latter are carried out at macroscopic scales. The omnipresence of dispersion forces at microscopic scales indicates the application of our results to other disordered materials too.

cond-mat.stat-mech↗

arXiv2021

Universality of Ultrasonic attenuation in amorphous systems at low temperatures

The competition between unretarded dispersion interactions between molecules prevailing at medium range order length scales and their phonon induced coupling at larger scales leads to appearance of nano-scale sub structures in amorphous systems. The complexity of intermolecular interactions gives rise to randomization of their operators. Based on a random matrix modelling of the Hamiltonian and its linear response to an external strain field, we show that the ultrasonic attenuation coefficient can be expressed as a ratio of two crucial length-scales related to molecular dynamics. A constant value of the ratio for a wide range of materials then provides a theoretical explanation of the experimentally observed universality of the ultrasonic attenuation coefficient at low temperatures.

cond-mat.soft↗

arXiv2021

Low temperature Universalities in amorphous systems: role of microscopic length scales

We find that a competition between dispersion forces among molecules in solids and their phonon mediated coupling leads to a natural length scale based on molecular parameters and relevant to decipher glass anomalies. For amorphous systems, the length scale is of the medium range topological orders and its ratio with the distance of closest approach between molecules turns out to be a constant. This in turn leads to a material independent, constant value of the ratio ${γ_l \over γ_t}$, with $γ_l$ and $γ_t$ as the coupling-strength for two amorphous molecules mediated by longitudinal and transverse phonons (also referred as Meissner-Berret ratio) and thereby provides a theoretical explanation of their experimentally obserevd quantitative universality in \cite{mb}. The above length scales are also related to Ioffe-Regel frequency and boson peak frequency of the vibrational spectrum and indicate that the former is of the same order as the latter for transverse phonon-dynamics.

cond-mat.stat-mech↗

arXiv2020

Spectral and Strength Statistics of Chiral Brownian Ensemble

Multi-parametric chiral random matrix ensembles are important tools to analyze the statistical behavior of generic complex systems with chiral symmetry. A recent study \cite{psmulti} of the former maps them to the chiral Brownian ensemble (Ch-BE) that appears as a non-equilibrium state of a single parametric crossover between two stationary chiral Hermitian ensembles. This motivates us to pursue a detailed statistical investigation of the spectral and strength fluctuations of the Ch-BE, with a focus on their behavior near zero energy region. The information can then be used for a wide range of complex systems with chiral symmetry. Our analysis also reveals connections of Ch-BE to generalized Calogero Sutherland Hamiltonian (CSH) and Wishart ensembles. This along with already known connections of complex systems without chirality to CSH strongly hints the later to be the "backbone" Hamiltonian governing the spectral dynamics of Complex systems.

cond-mat.stat-mech↗

arXiv2020

Extensive Nature of Long Range Interactions: Role of Disorder

The omnipresent disorder in physical systems makes it imperative to investigate its effect on the spatial range of interactions for which system remains thermodynamically extensive. Previously known bounds on the statistical free energy for clean systems \cite{fish} indicate it to be extensive only for the spatially short range interactions (decaying faster than $r^{-d}$ at large distance $r$ with $d$ as system dimension). We analyze the bounds for quantum systems with different types of disordered many body potentials e.g annealed, quenched, Gaussian or power law distributed. Our results indicate the dependence of the bounds on the multiple distribution parameters representing the potential which in turn permits, in contrast to clean potentials, more freedom to achieve the extensive limits even for arbitrary spatial ranges of the interactions.

cond-mat.stat-mech↗

arXiv2020

Spectral statistics of multi-parametric Gaussian ensembles with chiral symmetry

The statistics of chiral matrix ensembles with uncorrelated but multivariate Gaussian distributed elements is intuitively expected to be driven by many parameters. Contrary to intuition, however, our theoretical analysis reveals the existence of a single parameter, a function of all ensemble parameters, which governs the dynamics of spectral statistics. The analysis not only extends the similar formulation (known as complexity parameter formulation) for the Hermitian ensembles without chirality to those with it but also reveals the underlying connection between chiral complex systems with seemingly different system conditions as well as to other complex systems e.g. multi-parametric Wishart ensembles as well as generalized Calogero Sutherland Hamiltonian (CSH).

cond-mat.stat-mech↗