Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

434 alasdair urquhartThe theoretical developments of the 1970sand1980s were truly remarkable,and led to the development of a theory, in which the replica method played astarring role. The key papers, as well as some very useful background material,are collected in the volume (Mézard et al., 1987).Before describing what the replica method is, let me give a brief sketchof the SK model. The model consists of a very large number N of sites,each of which has a binary (classical) spin variable (taking the values +1and −1) associated with it. Each of the edges linking the sites is assigned arandom value; for simplicity let’s suppose that we randomly and independentlychoose the values +1 and−1 for the edges (in the SK model, the edgedistribution is Gaussian). In addition, let’s say that a positive edge representsfriendship and a negative edge enmity. Our problem then is to assign thespins into two camps so that we minimize conflicts—an administrator’sproblem. If the administrator can divide the people (spins) into two groupsso that all the people in one camp are friends, and are enemies of all thepeople in the other camp, then there is no conflict. But in general, thisis impossible, and we have what the physicists (and administrators!) callfrustration.How well can the administrator do in general? Suppose that we have 10,000people. The physicists, by using their mysterious non-rigorous methods, havecomputed that on average, the administrator cannot do much better thana situation in which the average person has thirty-eight fewer enemies thanfriends in their camp (Mézard et al., 1987,p.2). This is not a great improvementon the situation resulting from the administrator throwing up her hands, andsplitting the population in two by tossing a coin.We are dealing with what is basically a problem in finite combinatorics,like the satisfiability problem. The physicists, though, do not proceed viacombinatorics. They translate the problem into the form of a statisticalmechanical model, including a parameter called ‘temperature’, and concentratetheir efforts on measuring a crucial quantity called the free energy, representingthe degree of frustration in the system. Here is where the notorious replicamethod makes its appearance.The physicists make use of a certain expression for the logarithm of a physicalquantity, which involves letting a variable n tend to zero. They use the wellknown classical identityZ n − 1log Z = lim ,n→0 nwhere Z represents the partition function of the system. This is not puzzling initself, but their interpretation is. They interpret n as the number of replicas of