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

Non-intersecting Brownian Interfaces and Wishart Random Matrices