Moments of the 2D Directed Polymer in the Subcritical Regime and a Generalisation of the Erdös–Taylor Theorem

Abstract

AbstractWe compute the limit of the moments of the partition function $$Z_{N}^{\beta _N} $$ Z N β N of the directed polymer in dimension $$d=2$$ d = 2 in the subcritical regime, i.e. when the inverse temperature is scaled as $$\beta _N \sim \hat{\beta } \sqrt{\tfrac{\pi }{\log N}}$$ β N ∼ β ^ π log N for $$\hat{\beta } \in (0,1)$$ β ^ ∈ ( 0 , 1 ) . In particular, we establish that for every $$h \in {\mathbb {R}}$$ h ∈ R , $$ \lim _{N \rightarrow \infty } {{\mathbb {E}}} \big [\big (Z_{N}^{\beta _N}\big )^h \big ]=\big (\frac{1}{1-\hat{\beta }^2}\big )^{\frac{h(h-1)}{2}}.$$ lim N → ∞ E [ ( Z N β N ) h ] = ( 1 1 - β ^ 2 ) h ( h - 1 ) 2 . We also identify the limit of the moments of the averaged field $$\tfrac{\sqrt{\log N}}{N} \sum _{x \in {\mathbb {Z}}^2} \varphi (\tfrac{x}{\sqrt{N}})\big (Z_{N}^{\beta _N} (x)-1 \big )$$ log N N ∑ x ∈ Z 2 φ ( x N ) ( Z N β N ( x ) - 1 ) , for $$\varphi \in C_c({\mathbb {R}}^2)$$ φ ∈ C c ( 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 $$\begin{aligned} \frac{\pi }{\log N}\sum _{1 \le i<j\le h} {\textsf{L}}_N^{(i,j)}\xrightarrow [N \rightarrow \infty ]{(d)} \Gamma \big ( \tfrac{h(h-1)}{2},1\big ) , \end{aligned}$$ π log N ∑ 1 ≤ i < j ≤ h L N ( i , j ) → N → ∞ ( d ) Γ ( h ( h - 1 ) 2 , 1 ) , where $${{\textsf{L}}}^{(i,j)}_N$$ L N ( i , j ) denotes the collision local time by time N between copies i, j and $$\Gamma $$ Γ denotes the Gamma distribution. This generalises a classical result of Erdös and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960).