Pattern theorems

  • Neal Madras
  • Gordon Slade
Part of the Probability and Its Applications book series (PA)


In this chapter we shall prove a useful theorem due to Kesten (1963) about the occurrence of patterns on self-avoiding walks, and investigate a number of its applications. Briefly, a pattern is a (short) self-avoiding walk that occurs as part of a longer self-avoiding walk. Kesten’s Pattern Theorem says that if a given pattern can possibly occur several times on a self-avoiding walk, then it must occur at least a N times on almost all N-step self-avoiding walks, for some a > 0 (in this context, “almost all” means “except for an exponentially small fraction”). This can be viewed as a weak analogue of classical “large deviations” estimates for the strong law of large numbers, which say that certain events have exponentially small probabilities [see for example Chapter 1 of Ellis (1985)].


Triangular Lattice Outer Point Step Walk Dimensional Cube Unit Outer Normal Vector 
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Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Neal Madras
    • 1
  • Gordon Slade
    • 2
  1. 1.Department of Mathematics and StatisticsYork UniversityDownsviewCanada
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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