Random Path Representations

Abstract

The random path representation is a reformulation of Euclidean field theory. For scalar fields this representation exhibits the Schwinger functions in terms of a statistical distribution of paths; the effects of interaction are described by the intersection of paths. This representation was introduced by K. Symanzik for the investigation of Euclidean φ⁴ models [Symanzik, 1964, 1969]. Since the precise mathematical formulation of this work was beyond the scope of contemporary probability theory, the representation did not receive immediate attention. Eventually, however, random walk variants of this representation proved very fruitful in the study of lattice field theories. Random walk representions have been used to establish and to improve correlation inequalities such as those of Chapter 4. These inequalities yield a striking nonexistence theorem for continuum limits of lattice φ⁴ theories in dimension d ≥ 5 [Brydges, Fröhlich and Spencer, 1982; Aizenman, 1982; Fröhlich, 1982; Brydges, 1982]. They are the strongest indications to date of nonexistence for pure φ⁴ models in d = 4. The methods used in these investigations also relate to the local time picture of random walks [Dynkin, 1983; Fröhlich, 1984] and to the random walk approximations to Brownian motion.