Mutually avoiding paths in random media and largest eigenvalues of random matrices

Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently to the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution. The latter describes the fluctuations of the largest eigenvalue of a random matrix, drawn from the Gaussian unitary ensemble (GUE), and the result holds for a DP with fixed end points, i.e., for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of $N>1$ noncrossing continuum DP in a random potential, to the sum of the $N$th largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDFs and showing that they coincide for arbitrary $N$.

Figure 1

Sketch of the $N=2$ case: Two polymers start and end at (almost) coinciding points. The blue and yellow lines represent the point with maximal probability at any time slice $t$ and the disorder realization is chosen to have well-separated paths.

Figure 2

The empirical distribution $P_2\left(\gamma \right)$ for the rescaled sum $\gamma =\left({\lambda }_1+{\lambda }_2-\sqrt{8\mathcal{N}}\right)\sqrt{2}{\mathcal{N}}^{1/6}$ of the first $N=2$ eigenvalues in ${10}^5$ realizations of GUE matrices of size $\mathcal{N}=250$. The continuous lines are two different approximations for the tail obtained: from the inverse Laplace transformation of Eq. (12) (orange and continuous line); from the simple approximation $R_N\left(\gamma \right)=1$ in Eq. (16) (blue and dashed line).