By H W Szego, G P Kuhn

ISBN-10: 072042044X

ISBN-13: 9780720420449

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Additional info for Differential games and related topics

Example text

A discrete-time random walk in a random environment can be defined by Qeneral i z i n g the P~lya l a t t i c e random walk discussed in section 2, with the t r a n s i t i o n probability p(~ - #') replaced by a function for a transition p(~l~') . from site For each I ' at random from a space of allowed functions of I ' to s i t e , the function ~ . ~ in Eq. 1) p(~l~') is drawn The simplest example, and the one which has been most studied by mathematicians, is the one-dimensional walk with p(tl~') here {~£} = ~, 6£,£,+1 + {1 - a £ , } ~ £ , £ , _ 1 i s a set of independent, i d e n t i c a l l y assuming values in the i n t e r v a l [0,i].

N u m e r i c a l l y q u i t e accurate o u t s i d e of a small region (the " c r i t i c a l it It is r e q i o n " ) and does p r e d i c t a p e r c o l a t i o n t h r e s h o l d , a v i r t u e which is not shared by o t h e r a p p r o x i m a t i o n s based on f i n i t e it t = 1 . expansions in powers of has r e c e n t l y been shown by Sahimi et a l . p or 1 - p . ) The e f f e c t i v e medium a p p r o x i m a t i o n in i t s f o r the Bethe l a t t i c e , s i m p l e s t form is h i g h l y i n a c c u r a t e but an ingeneous m o d i f i c a t i o n of i t [160] enables the c o r r e c t c r i t i c a l due to Essam et a l .

8)] and v [Eq. 9)] e s t i mated by analysis of series expansions [91] for dimension E = 2,3 (but note the discussion following Eq. 15) below, and [99]). S(I') . is subject to the constraint that i t be a positive integer, the correspondence to self-avoidinq walks comes about by treating parameter and taking the formal l i m i t lysis. ~n . Generating functions for c n n as a continuous n ÷ 0 at an appropriate point in the anaand other s t a t i s t i c a l properties of s e l f - avoiding walks correspond to various thermodynamic properties of the n-vector model in the n ÷ 0 avoiding walks of ÷ ÷!