By F (Ed) Harary

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26. We use a Fourier analysis of functions f : Define the inner product by f, g : f, g = φ( f g), → R. → R, where φ = φ 1 , so that the L 2 -norm of f is given by 2 2 f φ( f 2 ) = = f, f . We call f Boolean if it takes values in the set {0, 1}. Boolean functions are in one–one correspondence with the power set of E via the relation f = 1 A ↔ A. 38) f 2 2 = φ( f 2 ) = φ( f ) = φ( A). For F ⊆ E, let u F (ω) = e∈F (−1)ω(e) = (−1) e∈F ω(e) , ω∈ . It can be checked that the functions u F , F ⊆ E, form an orthonormal basis for the function space.

Let = {0, 1} F where F is finite, and let P be a probability measure on c G. R. 6] Exercises 31 , and A ⊆ . Show that P( A) may be expressed as a linear combination of certain P( Ai ) where the Ai are increasing events. 3. (continuation) Let G = (V, E) be an infinite graph with finite vertexdegrees, and = {0, 1} E . An event A in the product σ -field of is called a cylinder event if it has the form A F × {0, 1} F for some A F ⊆ {0, 1} F and some finite F ⊆ E. Show that a sequence (µn ) of probability measures converges weakly if µn ( A) converges for every increasing cylinder event A.

Explain how Wilson’s algorithm may be adapted to sample uniformly from FW . d c G. R. Grimmett 6 February 2009 3 Percolation and Self-Avoiding Walk The central feature of the percolation model is the phase transition. The existence of the point of transition is proved by path-counting and planar duality. Basic facts about self-avoiding walks, oriented percolation, and the coupling of models are reviewed. 1 Phase transition Percolation is the fundamental stochastic model for spatial disorder. In its simplest form introduced in [47]1 , it inhabits a (crystalline) lattice and possesses the maximum of (statistical) independence.

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