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Summary For the simple random walk in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Z}^2$ \end{document} we study those points which are visited an unusually large number of times, and provide a new proof of the Erdős-Taylor Conjecture describing the number of visits to the most visited point.

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We show that as processes in (c, d, t) ∈ C(R 2 × R + 1)

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{\int_c^d {(L_t^{x + h} - L_t^x )^2 dx - 4h} \int_c^d {L_t^x dx} }} {{h^{3/2} }}\mathop \Rightarrow \limits^\mathcal{L} \left( {\frac{{64}} {3}} \right)^{1/2} \int_c^d {L_t^x d\eta (x)}$$ \end{document}
as h → 0 for Brownian local time L t x . Here η(x) is an independent two-sided Brownian motion.

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