The universality of the Tracy-Widom distribution could originate from a phase transition

Physicists from our lab have recently proposed a mechanism explaining the emergence of the Tracy-Widom distribution in a wide variety of problems in physics and mathematics. They showed that this distribution is associated to the critical behavior of a system of interacting particles, in the vicinity of a third order phase transition. The universality of this distribution would thus be inherited from the expected universality of critical physical systems.

It is well known that, in large systems, probability distributions describing the fluctuations of macroscopic observables often converge to limiting laws. The most famous one is certainly the Gaussian distribution which describes the fluctuations of the sum of independent random variables (of finite variance), but this is not the only one. During the last twenty years, there has been an important activity, both in mathematics and in physics, concerning another universal law: the Tracy-Widom distribution. It was indeed found in a wide variety of systems ranging from the longest increasing subsequence of random permutations, random growth processes and gauge theories (e.g., two-dimensional Yang-Mills theory) to finance. Despite this avalanche of results, the origin of the universality of this law remained a puzzle. By studying a system of particles interacting via a logarithmic potential, two physicists from our lab have recently shown that in such a system, the emergence of the Tracy-Widom distribution is associated to a critical behavior, close to a phase transition. This result reinforces strongly the conjecture that, in general, the Tracy-Widom distribution is associated to the critical fluctuations of certain types of phase transitions. This work was published in Journal of Statistical Mechanics : Theory and Experiment [1].

To understand this phenomenon, the researchers studied the largest eigenvalues of random matrices, whose typical fluctuations are precisely described by the Tracy-Widom distribution. They considered real symmetric or complex Hermitian NxN random matrices whose entries are independent Gaussian random variables, of zero mean and variance 1/N. Thus the average value of the largest eigenvalue has a finite value (=sqrt(2)) when N goes to infinity. Before their work, most of the studies had focused on the typical fluctuations of the largest eigenvalue, of order N^(-2/3) and thus very small, in the vicinity of its mean value sqrt(2). The key idea was to focus, not on the typical fluctuations, which are described by the Tracy-Widom distribution, but on large and rare fluctuations, far away from the mean value. The study of these large deviations can be mapped onto the study of a one-dimensional systems of particles interacting via logarithmic interactions, confined by a harmonic well centered at the origin, and in presence of an impenetrable wall. This wall enforces the particles to stay to its left. When the position of this wall crosses the mean value of the largest eigenvalue at sqrt(2), the system exhibits a transition between a weak coupling phase (when the wall is at the right of sqrt(2)) and a strong coupling phase (when the wall stays at the left of sqrt(2)). They demonstrated that this transition is a third order phase transition, where the third derivative of the free energy (with respect to the position of the wall) exhibits a discontinuity. The authors showed that the critical behavior of this system, close to the transition point, is precisely described by the Tracy-Widom distribution, implying the universality of this distribution.

These results have been recently highlighted in an article by M.  Buchanan in Nature Physics [2] as well as in an article in the online magazine Quanta, from the Simons-fondation [3], and more recently by the CNRS-Institut National de Physique [4].

Reference:

[1] S. N. Majumdar and G. Schehr, Top eigenvalue of a random matrix: large deviations and
third order phase transition, J. Stat. Mech. P01012 (2014).

[2] M. Buchanan, Equivalence Principle, Nature Phys.  10, 543 (2014).

[3] N. Wolchover, At the far ends of a new universal law, Quanta Magazine (October, 2014).

[4] L'universalitŽé de la distribution de Tracy-Widom proviendrait d'une transition de phase.