By Gordon Slade, Jean Picard
ISBN-10: 3540311890
ISBN-13: 9783540311898
ISBN-10: 3540355189
ISBN-13: 9783540355182
The lace growth is a strong and versatile technique for realizing the serious scaling of a number of types of curiosity in chance, statistical mechanics, and combinatorics, above their higher severe dimensions. those versions comprise the self-avoiding stroll, lattice bushes and lattice animals, percolation, orientated percolation, and the touch procedure. This quantity offers a unified and wide evaluate of the lace enlargement and its functions to those versions. effects comprise proofs of life of serious exponents and building of scaling limits. frequently, the scaling restrict is defined by way of super-Brownian movement.
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Additional resources for The lace expansion and its applications: Ecole d'Ete de Probabilites de Saint-Flour XXXIV, 2004
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For all with at most m points. Then disc Prnnf DISCREPANCY I 41 Usually, theorems about discrepancy are actually theorems about hereditary discrepancy. " For example, in Lecture 4 we showed that any family of n sets on n points has discrepancy at most [2« In (2n)] 1/2 . COROLLARY. If sd consists of n sets of arbitrary size Straight probabilistic methods could not achieve this result. If a set has size x then a random coloring gives discrepancy about x 1/2 which can be arbitrarily large. Combining linear algebra with the probabilistic method is very powerful.
That is, we have found specific yl, • • •, yn for which Fixing Xj with XjRj = \ R,•• , we complete the algorithm. 48 LECTURE 6 A parallel algorithm. Now we outline an efficient parallel algorithm to find x f , Vj with Note this is somewhat weaker than the « 3 / 2 V2/7r guaranteed above. LEMMA. 5. Set R = Y\ + • • • + Yn. Then Here is an argument due to Bonnie Berger. We calculate We're using here, for example, that for distinct /, j, k, I the expectation of the product YjYjYkYi is (by 4-wise independence) the product of the expectations, which is zero.
Whenp(n)« l/n we can characterize those p ( n ) for which a Zero-One Law holds. THEOREM. Let k be an arbitrary positive integer and assume Then for any first order statement A Proo/ We imitate the proof techniques used for p constant. Consider the statements: B: There are no k + 2 points containing a spanning tree; C: There is no cycle on
The lace expansion and its applications: Ecole d'Ete de Probabilites de Saint-Flour XXXIV, 2004 by Gordon Slade, Jean Picard
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