Summary of Lecture 1 ( Nov . 14 , 2008 ) Scribe : Radha Krishna Ganti
نویسنده
چکیده
2 Basic Techniques Definition 1 (Increasing event) A RV X is increasing on (Ω,F) if X(ω) 6 X(ω′) whenever ω 6 ω′. it is decreasing if −X is increasing. An event A ∈ F is increasing whenever its indicator function is an increasing variable, i.e., if 1A(ω) 6 1A(ω) whenever ω 6 ω′. Example: 1. Increasing event: The event A(x, y) that there exists an open path joining x to y. 2. Increasing RV: The number N(x, y) of the number of different open paths between x and y. If A is an event on the probability space for bond percolation, then if A is an increasing event, Pp(A) 6 Pp′(A) whenever p 6 p′. 2.1 The FKG Inequality Named after Fortuin, Kasteleyn, and Ginebre. This was first proved by Harris in 1960. Expresses the fact that increasing events can only be positively correlated.
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Summary of Lecture 17 - Nov . 10 , 2008 Scribe : David Tisza
1 Executive Summary We have continued with a review on the bond percolation and the introduced the coupling of the bond percolation process, then defined the critical phenomenon and the percolation probability. There is a theorem about the percolation probability is a non decreasing function and we bounded it between 1/3 and 2/3 for the two dimensional case. 2 Bond Percolation 2.1 Preliminaries...
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