We study the Laplacian-∞ path as an extreme case of the Laplacian-α random walk. Although, in the finite α case, there is reason to believe that the process converges to SLEκ, with κ = 6/(2α+ 1), we show that this is not the case when α = ∞. In fact, the scaling limit depends heavily on the lattice structure, and is not conformal (or even rotational) invariant.

We construct a Hamiltonian which in a scaling limit becomes equivalent to one that can be diagonalized by a Bogoliubov transformation. There may appear simultaneously a mean-field and a superconducting phase. They influence each other in a complicated way. For instance, an attractive mean field may stimulate the superconducting phase and a repulsive one may destroy it.

We consider two species of particles performing random walks in a domain in R with reflecting boundary conditions, which annihilate on contact. In addition there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species annihilate each other, particles of each species, chosen at random, give birth. We assume initially equa...