Non-intersecting Brownian Interfaces and Wishart Random Matrices

Abstract

We study a system of $N$ non-intersecting $(1+1)$-dimensional fluctuating elastic interfaces (`vicious bridges') at thermal equilibrium, each subject to periodic boundary condition in the longitudinal direction and in presence of a substrate that induces an external confining potential for each interface. We show that, for a large system and with an appropriate choice of the external confining potential, the joint distribution of the heights of the $N$ non-intersecting interfaces at a fixed point on the substrate can be mapped to the joint distribution of the eigenvalues of a Wishart matrix of size $N$ with complex entries (Dyson index $\beta =2$), thus providing a physical realization of the Wishart matrix. Exploiting this analogy to random matrix, we calculate analytically (i) the average density of states of the interfaces (ii) the height distribution of the uppermost and lowermost interfaces (extrema) and (iii) the asymptotic (large $N$) distribution of the center of mass of the interfaces. In the last case, we show that the probability density of the center of mass has an essential singularity around its peak which is shown to be a direct consequence of a phase transition in an associated Coulomb gas problem.

31 pages, 9 figures

References in corpus (12)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• Exact distribution of the maximal height of p vicious walkers
• Phase transitions of bipartite entanglement
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Nonintersecting Brownian excursions
• Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Maximum distributions of bridges of noncolliding Brownian paths

Cited by in corpus (51)

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• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• The Index Distribution of Gaussian Random Matrices
• Noninteracting fermions in a trap and random matrix theory
• How many eigenvalues of a Gaussian random matrix are positive?
• Probability distributions of Linear Statistics in Chaotic Cavities and associated phase transitions
• Living at the Edge: A Large Deviations Approach to the Outage MIMO Capacity
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Distribution of the time at which N vicious walkers reach their maximal height
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Dysonian dynamics of the Ginibre ensemble
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• A shortcut through the Coulomb gas method for spectral linear statistics on random matrices
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities
• Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Truncated linear statistics associated with the top eigenvalues of random matrices
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• On the concentration of large deviations for fat tailed distributions, with application to financial data
• Index Distribution of Cauchy Random Matrices
• Non-intersecting Brownian bridges in the flat-to-flat geometry
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Distribution of spectral linear statistics on random matrices beyond the large deviation function -- Wigner time delay in multichannel disordered wires
• Non-interacting fermions in hard-edge potentials
• Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
• Purity distribution for generalized random Bures mixed states
• Universal shocks in the Wishart random matrix ensemble - 1
• O'Connell's process as a vicious Brownian motion
• Free fermions and $\alpha$-determinantal processes
• Vicious Lévy flights
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Maximal distance travelled by N vicious walkers till their survival
• Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• General truncated linear statistics for the top eigenvalues of random matrices
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Bessel process, Schramm-Loewner evolution, and Dyson model
• Tails of Random Matrix Diagonal Elements: The Case of the Wishart Inverse
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"