  We consider the dynamics of classical particles or defects moving in a
fluctuating two-dimensional magnetic medium made of Ising spins. These defects
occupy empty sites, and each of them can move according to simple rules, by
exchanging its location with one of the neighboring or distant spin if the
energy is favorable, conserving the magnetization. We use a fermionic
representation of the theory in order to map the partition function into an
integral over Grassmannian variables. This model of annealed disorder can be
described by a Grassmannian action containing quartic interaction terms. We
study the critical behavior of this system as well as the entropy, specific
heat, and residual correlation functions which are evaluated within this
Grassmannian formalism. We found in particular that the correlations are
strongly attractive at short distances in the low temperature regime and for a
broader range of distances near the spin critical regime, and slightly
repulsive at large distances. These results are compared with Monte-Carlo
simulations.
