Background and notation

In this section we review briefly the basic facts and conjectures about the SAW that will be used in the remainder of this chapter. A comprehensive survey of the SAW, with emphasis on rigorous mathematical results, can be found in the excellent new book by Madras and Slade.  

Real polymers live in spatial dimension d= (ordinary polymer solutions) or in some cases in / = 2 (polymer monolayers confined to an interface ). Nevertheless, it is of great conceptual value to define and study the mathematical models— in particular, the SAW— in a general dimension d. This permits us to distinguish clearly between the general features of polymer behavior (in any dimension) and the special features of polymers in dimension d= 3. The use of arbitrary dimensionality also makes available to theorists some useful technical tools (e.g., dimensional regularization) and some valuable approximation schemes (e.g., expansion in / = 4 — e).  

First we define the quantities relating to the number (or entropy ) of SAWs Let cn (resp. c (x)) be the number of JV-step SAWs on starting at the origin and ending anywhere (resp. ending at jc). Then cjv and cat(jc) are believed to have the asymptotic behavior 

Next we define several measures of the size of an A-step SAW  

We then consider the mean values R])ff, RDj and R )j in the probability distribution which gives equal wei t to each iV-step SAW. Very little has been proven rigorously about these mean values, but they are believed to have the asymptotic behavior 

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