arXiv subjects
Publications and source records attributed to Yvan Saint-Aubin.
We introduce three non-local observables for the two-dimensional Ising model. At criticality, conformal field theory may be used to obtain theoretical predictions for their behavior. These formulae are explicit enough to show that their asymptotics are described by highest weights $h_{pq}$ from the Kac table for c=1/2 distinct from those of the three unitary representations (0, 1/16 and 1/2).
Langlands recently constructed a map that factorizes the partition function of a free boson on a cylinder with boundary condition given by two arbitrary functions in the form of a scalar product of boundary states. We rewrite these boundary states in a compact form, getting rid of technical assumptions necessary in his construction. This simpler form allows us to show explicitly that the map between boundary conditions and states commutes with conformal transformations preserving the boundary and the reality condition on the scalar field.
Probabilities of crossing on same-spin clusters, seen as order parameters, have been introduced recently for the critical 2d Ising model by Langlands, Lewis and Saint-Aubin. We extend Cardy's ideas, introduced for percolation, to obtain an ordinary differential equation of order 6 for the horizontal crossing probability pih. Due to the identity pih(r)+pih(1/r)=1, the function pih must lie in a 3-dimensional subspace. New measurements of pih are made for 40 values of the aspect ratio r (r in [0.1443,6.928]). These data are more precise than those obtained by Langlands et al as the 95%-confidence interval is brought to 4x10^{-4}. A 3-parameter fit using these new data determines the solution of the differential equation. The largest gap between this solution and the 40 data is smaller than 4x10^{-4}. The probability pihv of simultaneous horizontal and vertical crossings is also treated.
The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of conformal invariance and universality are established numerically.
The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It was rather to describe as concretely as possible, although in hypothetical form, the geometric aspects of universality, especially conformal invariance, in the context of percolation, and to present the numerical results that support the hypotheses. On the other hand, one ulterior purpose is to draw the attention of mathematicians to the mathematical problems posed by the physical notions. Some precise basic definitions are necessary simply to orient the reader. Moreover a brief description of scaling and universality on the one hand and of renormalization on the other is also essential in order to establish their physical importance and to clarify their mathematical content.