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Mondal T, Shukla P. Spectral statistics of multiparametric Gaussian ensembles with chiral symmetry. Phys Rev E 2020; 102:032131. [PMID: 33075878 DOI: 10.1103/physreve.102.032131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2020] [Accepted: 09/01/2020] [Indexed: 06/11/2023]
Abstract
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 formulation (known as complexity parameter formulation) for 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 between other complex systems, e.g., multiparametric Wishart ensembles as well as generalized Calogero-Sutherland Hamiltonians.
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Affiliation(s)
- Triparna Mondal
- Department of Physics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India
| | - Pragya Shukla
- Department of Physics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India
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Abstract
A Brownian ensemble appears as a nonequilibrium state of transition from one universality class of random matrix ensembles to another one. The parameter governing the transition is, in general, size-dependent, resulting in a rapid approach of the statistics, in infinite size limit, to one of the two universality classes. Our detailed analysis, however, reveals the appearance of a new scale-invariant spectral statistics, nonstationary along the spectrum, associated with multifractal eigenstates, and different from the two end-points if the transition parameter becomes size-independent. The number of such critical points during transition is governed by a competition between the average perturbation strength and the local spectral density. The results obtained here have applications to wide-ranging complex systems, e.g., those modeled by multiparametric Gaussian ensembles or column constrained ensembles.
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Affiliation(s)
- Suchetana Sadhukhan
- Department of Physics, Indian Institute of Technology, Kharagpur 721302, India
| | - Pragya Shukla
- Department of Physics, Indian Institute of Technology, Kharagpur 721302, India
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Kuczala A, Sharpee TO. Eigenvalue spectra of large correlated random matrices. Phys Rev E 2016; 94:050101. [PMID: 27967175 PMCID: PMC5161118 DOI: 10.1103/physreve.94.050101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2016] [Indexed: 11/07/2022]
Abstract
Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each other. The analytical results are confirmed by numerical simulations. The results have implications for the dynamics of neural and other biological networks where plasticity induces correlations in the connection strengths within the network. We find that the presence of correlations can have a major impact on network stability.
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Affiliation(s)
- Alexander Kuczala
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, California 92037, USA and Department of Physics, University of California, San Diego, California 92161, USA
| | - Tatyana O Sharpee
- Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, California 92037, USA and Department of Physics, University of California, San Diego, California 92161, USA
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Scaramazza JA, Shastry BS, Yuzbashyan EA. Integrable matrix theory: Level statistics. Phys Rev E 2016; 94:032106. [PMID: 27739706 DOI: 10.1103/physreve.94.032106] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2016] [Indexed: 06/06/2023]
Abstract
We study level statistics in ensembles of integrable N×N matrices linear in a real parameter x. The matrix H(x) is considered integrable if it has a prescribed number n>1 of linearly independent commuting partners H^{i}(x) (integrals of motion) [H(x),H^{i}(x)]=0, [H^{i}(x),H^{j}(x)]=0, for all x. In a recent work [Phys. Rev. E 93, 052114 (2016)2470-004510.1103/PhysRevE.93.052114], we developed a basis-independent construction of H(x) for any n from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the N→∞ limit provided n scales at least as logN; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values x=x_{0} or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at O(N^{-0.5}) deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest-neighbor level statistics.
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Affiliation(s)
- Jasen A Scaramazza
- Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
| | - B Sriram Shastry
- Physics Department, University of California, Santa Cruz, California 95064, USA
| | - Emil A Yuzbashyan
- Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
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Dey D, Shukla P. Universality classes in Coulomb blockade conductance peak-height statistics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052118. [PMID: 25493751 DOI: 10.1103/physreve.90.052118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2014] [Indexed: 06/04/2023]
Abstract
We investigate, using exact diagonalization techniques, the distribution of conductance peak heights in Coulomb blockade regime of a quantum dot connected to two leads under generic dot conditions. The study reveals a three-parametric dependence of the distribution: (i) two dot-lead contact characteristics and (ii) the complexity parameter, mimicking the combined effect of all dot conditions. This also indicates the presence of an infinite range of universality classes of conductance statistics, dominantly characterized just by the complexity parameter and global symmetry constraints.
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Affiliation(s)
- Dayasindhu Dey
- Department of Physics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India
| | - Pragya Shukla
- Department of Physics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India
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Shukla P. Thermodynamics of protein folding: a random matrix formulation. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:415106. [PMID: 21386596 DOI: 10.1088/0953-8984/22/41/415106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The process of protein folding from an unfolded state to a biologically active, folded conformation is governed by many parameters, e.g. the sequence of amino acids, intermolecular interactions, the solvent, temperature and chaperon molecules. Our study, based on random matrix modeling of the interactions, shows, however, that the evolution of the statistical measures, e.g. Gibbs free energy, heat capacity, and entropy, is single parametric. The information can explain the selection of specific folding pathways from an infinite number of possible ways as well as other folding characteristics observed in computer simulation studies.
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Affiliation(s)
- Pragya Shukla
- Department of Physics, Indian Institute of Technology, Kharagpur, India
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Dutta R, Shukla P. Criticality in the quantum kicked rotor with a smooth potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:031115. [PMID: 18851001 DOI: 10.1103/physreve.78.031115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2008] [Indexed: 05/26/2023]
Abstract
We investigate the possibility of an Anderson-type transition in the quantum kicked rotor with a smooth potential due to dynamical localization of the wave functions. Our results show the typical characteristics of a critical behavior, i.e., multifractal eigenfunctions and a scale-invariant level statistics at a critical kicking strength which classically corresponds to a mixed regime. This indicates the existence of a localization to delocalization transition in the quantum kicked rotor. Our study also reveals the possibility of other types of transition in the quantum kicked rotor, with a kicking strength well within the strongly chaotic regime. These transitions, driven by the breaking of exact symmetries, e.g., time reversal and parity, are similar to weak-localization transitions in disordered metals.
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Affiliation(s)
- Rina Dutta
- Department of Physics, Indian Institute of Technology, Kharagpur, India
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Dutta R, Shukla P. Complex systems with half-integer spins: symplectic ensembles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:051124. [PMID: 18233640 DOI: 10.1103/physreve.76.051124] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2007] [Indexed: 05/25/2023]
Abstract
We study the statistical behavior of the Hermitian operators of complex systems with half-integer angular momentum and time-reversal symmetry. The complexity leads to randomization of the operators which can then be modeled, following maximum entropy hypothesis, by multiparametric Gaussian ensembles of real-quaternion matrices. The modeling shows that it is possible to classify the statistical behavior of spin-based complex systems into a continuum of universality classes characterized by a single parameter which is a function of all system parameters.
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Affiliation(s)
- Rina Dutta
- Department of Physics, Indian Institute of Technology, Kharagpur, India
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Shukla P. Eigenfunction statistics of complex systems: a common mathematical formulation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:051113. [PMID: 17677028 DOI: 10.1103/physreve.75.051113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2006] [Indexed: 05/16/2023]
Abstract
We derive a common mathematical formulation for the eigenfunction statistics of Hermitian operators, represented by a multiparametric probability density. The system information in the formulation enters through two parameters only, namely, system size and the complexity parameter, a function of all system parameters including size. The behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling. The existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties. This also suggests the possibility to classify them in various universality classes defined by the complexity parameter.
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Affiliation(s)
- Pragya Shukla
- Department of Physics, Indian Institute of Technology, Kharagpur, India
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