Wigner Time-Delay Distribution in Chaotic Cavities and Freezing Transition

Using the joint distribution for proper time delays of a chaotic cavity derived by Brouwer, Frahm, and Beenakker [Phys. Rev. Lett. 78, 4737 (1997)], we obtain, in the limit of the large number of channels $N$, the large deviation function for the distribution of the Wigner time delay (the sum of proper times) by a Coulomb gas method. We show that the existence of a power law tail originates from narrow resonance contributions, related to a (second order) freezing transition in the Coulomb gas.

Figure 1

Sketch of the distribution of $s=N{\tau }_{\mathrm{W}}$. The dashed line at $s=s_N\simeq 1+(4/N{)}^{1/3}$ separates the two phases of the Coulomb gas with densities represented in the small figures on the left and the right, respectively.

Figure 2

The optimal density of eigenvalues for different values of $s$; when $s$ increases, the density eventually freezes to the MP law (dashed line).

Figure 3

Large deviation function ${\Phi }_{-}(s)$ (i.e., rescaled energy of the gas). The freezing transition takes place at $s_{\infty }=1$. The metastable branch terminates at $s_c=1.1738\dots$ Inset: Large deviation function ${\Phi }_{+}(s)$ (i.e., $1/N$ correction to the rescaled energy).