Theoretical
Physics(DTP)
Physique théorique (DPT)
Yvan SAINT-AUBIN
Université
de Montréal
Behavior
of the Two-Dimensional Ising Model at the Boundary of a Half-Infinite Cylinder*
The two-dimensional Ising model is studied at the boundary of a
half-infinite cylinder. The three regular lattices (square, triangular and
hexagonal) and the three regimes (sub-, super- and critical) are
discussed. The probability of having
precisely 2n spinflips at the boundary is computed as a function of the positions ki's, i = 1,...,2n, of the spinflips. The limit when the mesh goes to zero is
obtained. For the square lattice, the probability of having 2n
spinflips, independently of their position, is also computed. These results are
obtained as consequences of Onsager's solution and are rigorous.
In the special case of precisely 4 spinflips, we use conformal
field theory to give a prediction for the following probability. Let q 1, q 2, q 3 and q 4 be the positions of the flips along the
boundary. We
give the probability distribution that the contour leaving q 1 ends at q 2 instead than at q 4.
The behavior of this function when q 2 - q 2 ® 0 is
described by a power law
with an exponent 
that belongs to the Kac
table but that corresponds to a non-unitarizable highest-weight
representation. We check that this
prediction agrees with a Monte-Carlo simulation.
*In
collabaration with L.P. Arguin, Princeton University and H. Aurag, CAE.