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›› 2014, Vol. 31 ›› Issue (2): 145-154.DOI: 10.7523/jssn.2095-6134.2014.02.001

• Research Articles •     Next Articles

A note on winding angles for supercritical percolation

KONG Ruiyuan, GUO Tiande   

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • Received:2012-12-04 Revised:2013-04-24 Online:2014-03-15
  • Supported by:

    Supported by National Natural Science Foundation of China(71271204,11331012,and 11101420)

Abstract:

This paper studies the supercritical Bernoulli bond percolation in two dimensions, focusing on properties of the winding angles of open paths. Under the condition that there exists an open path from the origin to the boundary of box B(n), we prove that a law of large numbers holds for the maximum of such paths' winding angles. Moreover, we show that for any δ>0, there is a high probability that the distance between two adjacent contact points in two neighboring "innermost" left-right crossings is less than log1+δn for sufficiently large n.

Key words: winding angle, crossing, law of large numbers, contact point

CLC Number: