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Obituary
Figure 5. Percolation in a rectangle
Figure 6. An interface in a small domain
of black cells appear, and the outer boundaries of the black
clusters are inner boundaries of white clusters and vice-versa.
It is conjectured (and it is now proved in several important
cases) that such interfaces behave asymptotically in a conformally
invariant way on large scale (we will not detail here
what is meant by this). Combining this conjecture with the
previous analysis, Oded Schramm concluded that:
Key conclusion. For a large class of models from statistical
physics, if they behave in a conformally invariant way in the
scaling limit, then the laws of the interfaces converge (in the
limit when the mesh of the lattice vanishes) to that of SLE
curves. The value of κ depends on the studied model.
Study of SLE processes and consequences.
The definition of SLE processes makes it possible to compute
explicitly the probability of certain events. For example
(see Oded’s paper “A percolation formula” [10]), the probability
that an SLE process in a simply domain D from A ∈ ∂D
to B ∈ ∂D passes to the right of a given point Z ∈ D can be
expressed easily via a differential equation. More precisely,
one first maps the domain D conformally onto the upper halfplane
in such a way that A and B are respectively mapped
onto 0 and ∞, and the image of Z has height 1. The probability
in question therefore just depends on the real value x of
the image of Z. One can then study how this probability h(x)
evolves as the SLE starts to grow, and one is led to the differential
equation κh ′′ (x)+8x/(1 + x 2 )h ′ (x)=0 that enables
one to deduce that h is a hypergeometric function of the type
that had appeared in the conformal field theory literature.
More generally, and sometimes with some technical difficulties,
it is possible to compute the asymptotic behaviour of
certain exceptional events. Rohde and Schramm [9] have, for
instance, shown that when κ ≤ 8, the asymptotic probability
that an SLE of parameter κ passes through the ε-neighbourhood
of a given point z ∈ D decays like ε 1−κ/8 when ε → 0.
This implies that the Hausdorff dimension d(κ) of the curve is
not larger than 1 + κ/8 (in fact, d(κ) is equal to this value, as
later proved by Vincent Beffara). Such fractal dimensions are
closely related to the critical exponents studied by our friends
in the physics community via various techniques including
the above-mentioned conformal field theory.
The study of SLE processes has enabled mathematicians
to prove results for all the models for which asymptotical conformal
invariance is established. This includes simple random
walks, loop-erased random walks and uniform spanning trees,
the critical percolation model that we have briefly described
and the Ising model on the square grid (the conformal invariance
of these last two models has been proved recently by
Stas Smirnov). Here are two examples extracted from joint
papers of Oded with Greg Lawler and myself:
Figure 7. A planar Brownian motion
Theorem 7 ([11]). Consider a planar Brownian motion
(Bt,t ∈ [0,1]) and define its outer boundary F as the boundary
of the unbounded connected component of the complement of
B[0,1]. Then the Hausdorff dimension of F is almost surely
equal to 4/3.
Theorem 8 ([12]). Consider the percolation model when one
colours each cell of the honeycomb lattice independently black
or white with probability 1/2. The probability that there exists
a white path originating at the origin that reaches distance
R from the origin decays asymptotically like R −5/48+o(1) as
R → ∞.
SLE has now become a research area of its own. The following
recent (some are still being written up) important results
by Oded on the subject are not so easy to state without a long
introduction, so that we only describe them heuristically.
• In the spirit of his earlier papers on the subject with Itai
Benjamini and Gil Kalai [15], and with Jeff Steif [16],
Oded, together with Christophe Garban and Gábor Pete,
gives in [17] precise and complete answers to the question
of “sensitivity to small perturbations” of critical percolation
on a large scale. In another paper, they provide in [18]
a detailed description of percolation near its critical point.
12 EMS Newsletter March 2009