We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the Swiss cheese picture of Bolthausen, van den Berg and den Hollander [7].
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机译：我们研究了欧几里德格子上瞬态行走范围的边界的向下偏差。 我们描述了行走所采用的最佳策略，以缩小其范围的边界。 我们开发的技术同样适用于该范围，并为瑞士奶酪的Pathwise语句为Bolthausen，Van den Berg和Den Hollander [7]提供了瑞士奶酪图片[7]。
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