# Tracy-Widom universality for polymers in intermediate disorder

13 Feb 2018Jeremy Quastel and I started working on this project when I visited the Fields Institute in June 2014. The most recent version of the paper fixes some serious errors, but it’s still under review and we have not posted the updated version on the arxiv.

When I was a graduate student, I had been reading about the Lindeberg style argument that Terry Tao and Van Vu used in the random matrix universality papers from 2010. It’s a simple argument originally introduced by Lindeberg in 1922 to prove the central limit theorem. Davar Khosnevisan believes that the method is due to Lyapunov; I struggled through parts of Lyapunov’s original paper in French to see if I could verify this, but I couldn’t quite get through it. The method was strengthened with a truncation argument and repopularized by Trotter in 1959. More recently, it has been applied quite successfully to more modern models by Chatterjee (2005).

I was trying to apply the method to first-passage percolation, and mentioned this to Jeremy when I started at the Fields institute. But it’s clear that a naive perturbation like this is not going to work: if one tries to use the method to prove that the fluctuations of the passage time are universal, one necessarily proves that the time-constant should be universal as well. This is not generally true in last-passage percolation at least (see Cox-Durrett 1981 for a first-passage example), so at the moment, this does seem like a serious obstruction. Luckily, Jeremy had co-discovered the intermediate disorder regime in directed polymers a few years earlier, and he suggested that we try to apply it here.

The analog of the time-constant in the $d+1$ dimensional polymer model is the free-energy. There are two parameters here: the length of the polymer and the inverse temperature $\beta$. The fluctuations of the free energy are thought to be given by the Tracy-Widom GUE (TW) distrbution even in the intermediate disorder regime $O(1) \gg \beta \geq O(N^{-1/4})$. A non-rigorous argument for this appears in the physics paper by Alberts, Khanin and Quastel (2014). In the special regime when $\beta = \tilde\beta N^{-1/4}$, Alberts-Khanin-Quastel (2015) proved that the partition function of the polymer scales to the solution of the stochastic heat equation. The log of solution to the stochastic heat equation has long-time $(\tilde\beta \to \infty)$ TW fluctuations (Amir-Corwin-Quastel 2011). So in the special intermediate-disorder double limit $N \to \infty$ and then $\tilde\beta \to \infty$, it’s known quite generally that the polymer has TW fluctuations. However, when $O(1) \gg \beta \gg O(N^{-1/4})$, the centered and rescaled log partition function of the polymer is converge directly to a TW random variable without having to take this double limit. But this had not been proved for any standard discrete polymer.

The free-energy does indeed become “universal” in the intermediate disorder regime. So there is some hope of using the Lindeberg replacement strategy. Moreover, since $\beta \to 0$ in this regime at a given rate, we can use this to control the error in a Taylor expansion of the log partition function. For example, this $\beta \to 0$ property is implicit in the Sherrington-Kirkpatrick spin glass, and hence the Lindeberg strategy has been quite successful there.

The situation is complicated by the fact that no standard polymer is known to be in the TW universality class, and so an “invariance theorem” is a little unsatisfactory. Seppalainen’s log-gamma polymer is not quite a standard polymer, but it’s solvable and can be shown to have TW fluctuations, at least for $\beta \geq \beta* > 0$ (O’Connell-Seppalainen-Zygouras 2014, Borodin-Corwin-Remenik 2013, Borodin-Corwin-Ferrari-Veto 2015). We first show that the log-gamma polymer has TW fluctuations in intermediate disorder; then we use the Lindeberg strategy to prove that polymers that are close to the log-gamma polymer also have TW fluctuations.