For a symmetric random walk in Z2 with 2+δ moments, we represent |ℛ(n)|, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each k≥1
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\[(\log n)^{k}\Biggl[\frac{1}{n}|\mathcal{R}(n)|+\sum_{j=1}^{k}(-1)^{j}\biggl(\frac{1}{2\pi}\log n+c_{X}\biggr)^{-j}\gamma_{j,n}\Biggr]\to 0\qquad\mbox{a.s.,}\]
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where Wt is a Brownian motion, $W^{(n)}_{t}=W_{nt}/\sqrt{n}$ , γj,n is the renormalized intersection local time at time 1 for W(n) and cX is a constant depending on the distribution of the random walk.