My research
interest focuses on the Schramm-Loewner evolution (SLE for short), which
describes some random fractal curves in plane domains whose distribution is
preserved under conformal (analytic and one-to-one) maps. It was introduced by
Oded Schramm in 1999 to study the scaling limits of loop-erased random walk in
the plane. The definition combines the Charles Loewner’s equation (1923) from
Complex Analysis with a random input function. Loewner’s equation generates a
map from C([0,T],R) to the set of “curves” in plane domains. If the input
function is times
a standard Brownian motion then the random curve obtained is called SLE with
parameter κ; and we write SLE_{κ} to emphasize this parameter.

The merit of SLE is that one can now use the tools from Stochastic Analysis to
analyze some random curves in the plane. Such curves include the scaling limits
of many two dimensional statistical physics models, e.g., critical site
percolation, Ising models at critical temperature, Gaussian free field,
loop-erased random walk, and uniform spanning tree. These lattice models have
been proved to converge to SLE with different parameters. Another important
application of SLE is that it was used to prove Mandelbrot’s conjecture: the
boundary of plane Brownian motion has fractal dimension 4/3.

Of many variants of SLE, the chordal SLE and radial SLE are most well-known.
They are both defined in simply connected domains. A chordal SLE curve grows
from one boundary point to another boundary point; a radial SLE curve grows
from a boundary point to an interior point.

One of my research projects is to extend SLE to multiply connected domains, and
relate them to lattice models in these domains. I have defined SLE in doubly
connected domains by introducing a new kind of Loewner equation. Latter I
defined a family of SLE in multiply connected domains using the traditional
Loewner’s equation, and proved that they are the scaling limits of loop-erased
random walk in these domains. One may consider the scaling limits of other
lattice models in multiply connected domains.

My recent interest is the application of a new tool in the area of SLE: the
stochastic coupling technique. Roughly speaking, the coupling technique allows
two SLE curves to grow in the same plane domain simultaneously. If these two
curves satisfy that every point on one curve will be visited by the other, then
they overlap. The first application of this technique was to show that chordal
SLE_{κ} satisfies reversibility if κ≤4. This means that the chordal SLE_{κ}
curve from a to b is the same as the chordal SLE_{κ} curve from b to a.
This technique was also used to show the duality of SLE: the outer boundary of
an SLE_{κ} curve with κ>4, which is not a simple curve, has the
shape of an SLE_{16/κ }curve. Here the parameter 16/κ is the dual of κ.
It is known that SLE_{κ} and SLE_{16/κ} have the same *central
charge*. The coupling technique may also be used to study the reversal curve
of radial SLE. Another interesting application is that one may erase loops on a
plane Brownian motion in the order they appear to get a simple curve, which is
an SLE_{2} curve.

Here are a few pictures from the area of SLE.

**Figure 1: **The red part is a
plane Brownian motion in the disc (approximated by random walk on a square
lattice with very small mesh). The black curve is the outer boundary of this
Brownian motion. The boundary curve has fractal dimension 4/3, which was
conjectured by Mandelbrot, and proved by G.F. Lawler, O. Schramm, and W. Werner
using SLE. In fact, it is an SLE_{8/3} curve.

**Figure 2: **The hexagon faces on
the bottom lines are colored in such a way that the left half of them are
colored gray, and the right half of them are colored white. All other hexagons
are colored gray or white independently with probability 1/2. The red line runs
along the boundaries of the hexagons in such a way that the hexagons on its
left is gray, and the hexagons on its right is white. When the size of the
hexagons tends to 0, this red curve converges to SLE_{6} (by S.
Smirnov).

**Figure 3: **This zigzag curve on
the left is an SLE curve. The function g_{t} is a conformal map that
maps the upper half plane without the curve (up to time t) onto the whole upper
half plane. The curve is understood as a part of the boundary of the domain,
and g_{t} maps the two sides of the curve onto two real intervals. The
variable t is the time parameter of the curve. The family of maps {g_{t}}
satisfies Loewner's equation. The way that SLE people analyze the fractal curve
is to study the functions {g_{t}} instead.