Universality of directed polymers in the intermediate disorder regime

The directed polymer was introduced by Huse and Henley as a model for the domain wall in a ferromagnetic Ising model with random bond impurities. This model depends on a parameter $beta$, the inverse temperature. We consider the intermediate disorder regime, which consists in taking $beta$ to depend on the length of the polymer 2n, with $beta=n^{$ for some $alpha>0$. In this regime, there is a critical phase transition that happens at $alpha=1/4$. When $alpha > 1/4$, the fluctuations of the free energy are of order $n}{(1-4alpha)/4}$ and converge to a Gaussian. For $alpha < 1/4$, it was conjectured that the polymer should fall back in the Kardar—Parisi—Zhang universality class, and that the fluctuations should instead be of order $n^{(1-4alpha)/3}$, and converge after rescaling to the Tracy—Widom GUE distribution. In this talk, I will sketch a proof of this conjecture for $1/8 < alpha < 1/4$ for arbitrary i.i.d weights with exponential moments, using a kind of “local chaos expansion”.

• Speaker: Julian Ransford (Cambridge)
• Tuesday 15 October 2024, 14:00–15:00
• Venue: MR12.
• Series: Probability; organiser: Jason Miller.