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{tfrac{pi }{log N}} for hat{beta } in (0,1). In particular, we establish that for every h in {mathbb {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}}. 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 ), for varphi 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 πlogN∑1≤i<j≤hLN(i,j)→N→∞(d)Γ(h(h-1)2,1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\pi }{\\log N}\\sum _{1 \\le i<j\\le h} {\ extsf{L}}_N^{(i,j)}\\xrightarrow [N \\rightarrow \\infty ]{(d)} \\Gamma \\big ( \ frac{h(h-1)}{2},1\\big ) , \\end{aligned}$$\\end{document}where {{textsf{L}}}^{(i,j)}_N 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 and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960).