Moments of the 2d directed polymer in the subcritical regime and a generalisation of the Erdös-Taylor theorem

Abstract

We compute the limit of the moments of the partition function $Z_N^{{\beta }_N}$ of the directed polymer in dimension $d=2$ in the subcritical regime, i.e. when the inverse temperature is scaled as ${\beta }_N\sim \hat{\beta }\sqrt{\frac{\pi }{\log N}}$ for $\hat{\beta }\in (0,1)$. In particular, we establish that for every $h\in \mathbb{R}$, ${\lim }_{N\to \infty }\mathbb{E}\left[\right(Z_N^{{\beta }_N}{)}^h]=(\frac{1}{1-{\hat{\beta }}^2}{)}^{\frac{h(h-1)}{2}}.$ We also identify the limit of the moments of the averaged field $\frac{\sqrt{\log N}}{N}{\sum }_{x\in {\mathbb{Z}}^2}\phi (\frac{x}{\sqrt{N}})(Z_N^{{\beta }_N}(x)-1)$, for $\phi \in C_c({\mathbb{R}}^2)$, as those of a gaussian free field. As a byproduct, we identify the limiting probability distribution of the total pairwise collisions between $h$ independent, two dimensional random walks starting at the origin. In particular, we derive that $$\frac{\pi }{\log N}\sum _{1\le i<j\le h}{\mathsf{L}}_N^{(i,j)}\underset{N\to \infty }{\overset{(d)}{\to }}\Gamma (\frac{h(h-1)}{2},1),$$ where ${\mathsf{L}}_N^{(i,j)}$ denotes the collision local time by time $N$ between copies $i,j$ and $\Gamma$ denotes the Gamma distribution. This generalises a classical result of Erdös-Taylor.

arXiv commentRevised introduction and bibliography and some results are strengthened. 32 pages, 1 figure

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Cited by in corpus (3)

• The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos
• Gaussian Limits for Subcritical Chaos
• Two dimensional delta Bose gas in a weighted space