calculations indicate that the transverse displacement and free energy fluctuations of 1 + 1 dimensional directed random walks in media with sign randomness are characterized by the same exponents as directed walks in ordinary random media, with positive statistical weights.
Professor Michael E. Fisher has had a longstanding interest in the properties of random walks and their applications in statistical physics: see, for example, his delightful Boltzmann Medalist address, "Walks, walls, wetting, and melting" [ 1 ]. We hope he enjoys this small contribution to the subject.
Directed 1 + 1 dimensional random walks in random media are now well understood in many respects [ 2-5 ], while continuing to be of interest [6][7][8]. Such walks provide simple models of polymers in random media or interfaces in random-bond Ising models. They are characterized by a wandering exponent v= ~, which describes the scaling of the typical transverse displacement La_ with the longitudinal displacement L, via Ll ~ L I~', and by a fluctuation exponent o9 which describes the growth in the sample-to-sample fluctuations of the free energy (described below) with longitudinal displacement, via AF~ L~.
To make the discussion more concrete, consider a square lattice rotated clockwise by 45 Β°, as shown in fig. 1. Directed walks from the origin to points in the first quadrant propagate strictly to the right; a walk which has reached (x, y) can go to either (x+ 1, y) or (x, y+ 1 ) at the next step.
Associate each nearest-neighbor bond ~l of the lattice with a random value ot [r; r' ] drawn independently from some distribution. For the moment suppose that a~>0; this corresponds to the usual directed-random-walk-in-random-media problem, with cz corresponding to e -`/T and E being a random energy. The statistical weight of an individual walk is given by W(F) = l-I t,;,' l~r a [r; r' ], where F denotes the set of bonds comprising the walk. The partition function Z[r] is defined as the sum of the weights of all directed walks from the origin to r. This partition function depends implicitly on the particular realization of disorder; let us use angular brackets to denote averages over the or[r; r' ].
#~ Note that one could instead associate a random value with each site rather than each bond, but one expects the same asymptotic properties to be obtained.