You are currently browsing the tag archive for the ‘Stochastic Loewner equation’ tag.

I am very saddened (and stunned) to learn that Oded Schramm, who made fundamental contributions to conformal geometry, probability theory, and mathematical physics, died in a hiking accident this Monday, aged 46.  (I knew him as a fellow editor of the Journal of the American Mathematical Society, as well as for his mathematical research, of course.)  It is a loss of both a great mathematician and a great person.

One of Schramm’s most fundamental contributions to mathematics is the introduction of the stochastic Loewner equation (now sometimes called the Schramm-Loewner equation in his honour), together with his subsequent development of the theory of this equation with Greg Lawler and Wendelin Werner.  (This work has been recognised by a number of awards, including the Fields Medal in 2006 to Wendelin.)  This equation (which I state after the jump) describes, for each choice of a parameter $\kappa > 0$, a random (fractal) curve $SLE(\kappa)$ in the plane; this random curve can be viewed as a nonlinear variant of Brownian motion, although the SLE curves tend to cross themselves much less frequently than Brownian paths do.  By the nature of their construction, the $SLE(\kappa)$ curves are conformally invariant: any conformal transformation of an $SLE(\kappa)$ curve (respecting the boundary) gives another curve which has the same distribution as the original curve.  (Brownian motion is also conformally invariant; given the close connections between Brownian motion and harmonic functions, it is not surprising that this fact is closely related to the fact that the property of a function being harmonic in the plane is preserved under conformal transformations.) Conversely, one can show that any conformally invariant random curve distribution which obeys some additional regularity and locality axioms must be of the form $SLE(\kappa)$ for some $\kappa$.

The amazing fact is that many other natural processes for generating random curves in the plane – e.g. loop-erased random walk, the boundary of Brownian motion (also known as the “Brownian frontier”), or the limit of percolation on the triangular lattice – are known or conjectured to be distributed according to $SLE(\kappa)$ for some specific $\kappa$ (in the above three examples, $\kappa$ is 2, 8/3, and 6 respectively).  In particular, this implies that the highly non-trivial fact that such distributions are conformally invariant, a phenomenon that had been conjectured by physicists but which only obtained rigorous mathematical proof following the work of Schramm and his coauthors.

[Update, Sep 6: A memorial blog to Oded has been set up by his Microsoft Research group here.  See also these posts by Gil Kalai, Yuval Peres, and Luca Trevisan.]

### Recent Comments

 Terence Tao on 245C, Notes 2: The Fourier… Anonymous on A problem involving power… Anonymous on 245C, Notes 2: The Fourier… Georges Elencwajg on Math 246A, Notes 4: singularit… Anonymous on A problem involving power… Petrus3743 on A geometric proof of the impos… Anonymous on A problem involving power… maxbaroi on A problem involving power… elianto84 on A problem involving power… Mutual Knowledge vs.… on It ought to be common knowledg… Anonymous on A problem involving power… Terence Tao on A problem involving power… Anonymous on A problem involving power… Anonymous on A problem involving power… rsj on A problem involving power…