Review of the proof that 2d directed polymer and stochastic heat equation partition functions converge, in a critical window, to a unique object called the critical 2d stochastic heat flow.
On the Central Limit Theorem for the log-partition function of 2D directed polymers
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abstract
The log-partition function $ \log W_N(\beta)$ of the two-dimensional directed polymer in random environment is known to converge in distribution to a normal distribution when considering temperature in the subcritical regime $\beta=\beta_N=\hat{\beta}\sqrt{\pi/\log N}$, $\hat{\beta}\in (0,1)$ (Caravenna, Sun, Zygouras, Ann. Appl. Prob. (2017)). In this paper, we present an elementary proof of this result relying on a decoupling argument and the central limit theorem for sums of independent random variables. The argument is inspired by an analogy of the model to branching random walks.
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The Critical 2d Stochastic Heat Flow and Related Models
Review of the proof that 2d directed polymer and stochastic heat equation partition functions converge, in a critical window, to a unique object called the critical 2d stochastic heat flow.