Bharali, Gautham

Ramadas, T. R.

English

Publications of the IAS Fellows

PERSONAL NEWS 
 
CURRENT SCIENCE, VOL. 96, NO. 2, 25 JANUARY 2009 297
Oded Schramm and the Schramm–Loewner evolution: In memoriam 
 
Oded Schramm, the inventor of the sto-
chastic Loewner evolution (SLE), died 
on 1 September 2008 in a tragic hiking 
accident in the Cascades Mountains. He 
was forty-six. He is survived by his wife 
and two children. 
 Schramm left behind an impressive 
legacy of research in probability and sto-
chastic processes, statistical mechanics, 
and discrete and computational geo-
metry. He brought highly original mathe-
matical insights to the study of critical 
phenomena in lattice models in statistical 
physics. Schramm will be sorely missed 
by the mathematics and the statistical 
physics communities. 
 Oded Schramm was born in Jerusalem 
in December 1961. At the Hebrew Uni-
versity, he received a B Sc in mathemat-
ics and computer science, and an M Sc in 
mathematics under the direction of Gil 
Kalai. His Ph D (1990) was from Prince-
ton – his supervisor: William Thurston. 
After a postdoctoral stint at the Univer-
sity of California, San Diego, Schramm 
moved to the Weizmann Institute in 
1992. In 1999, he joined the Theory 
Group at Microsoft Research in Red-
mond. 
 Schramm was an avid hiker. He listed 
his interests – on his Microsoft webpage 
some years back – thus: ‘Percolation, 
two-dimensional random systems, criti-
cal systems, SLE, conformal mappings, 
dynamical random systems, discrete and 
coarse geometry, mountains’. 
Conformal invariance of scaling 
limits and the SLE 
The most celebrated of Schramm’s con-
tributions is his work on scaling limits of 
two-dimensional lattice models in statis-
tical mechanics. The goal of statistical 
mechanics, roughly speaking, is to study 
the thermodynamic properties of large 
ensembles of interacting particles. The 
dimension of the ‘configuration space’ 
associated with this ensemble is the 
number of its degrees of freedom. In the 
large-number limit, the configuration 
space is infinite-dimensional. In mathe-
matical terms, statistical mechanics is 
concerned with the construction of thermo-
dynamically significant probability 
measures on these infinite dimensional 
spaces. These measures are constructed 
as limits of measures on suitably trun-
cated finite-dimensional systems. Typi-
cally, there are only finitely many 
variables (dimensions) associated with a 
finite volume V – and the limit involves 
taking V → ∞ (‘infinite volume’) limit. 
The approximating measures depend on 
the interaction energy EV of the specific 
system, as well as on thermodynamic  
parameters (e.g. temperature T, magnetic 
field, etc.). 
 In two dimensions, the theory is parti-
cularly rich, and a variety of techniques – 
probabilistic, algebraic and heuristic – 
have been deployed to great effect, start-
ing with Onsager’s solution of the Ising 
model. We shall use this familiar model 
as a guide to some of the technical con-
structs that underpin the work of 
Schramm and his colleagues. To begin 
with, one considers a ‘square lattice’, 
namely the infinite grid Γa in R2. (The 
parameter a determines how fine a grid 
we get.) To each vertex v of the grid, one 
associates a variable (‘spin’) σv taking 
values 1 (denoting ‘up’ spin) or –1 (de-
noting ‘down’ spin). The ‘configuration 
space’ consists of all assignments of spin 
at every vertex. (One can view this space 
as an infinite cartesian product of 
{1, −1} indexed by the vertices v.) Given 
a finite region V, the function EV is de-
fined on the space ∏v∈V{1, −1} by the 
following formula: 
 
1 2( ) ( ) ,V v l v l
l V
E Jσ σ
∈
=∑  
where the sum runs over edges l of the 
grid contained in V, and vi(l) are the ver-
tices at either end of the edge l. Here J is 
a real parameter that measures how 
nearby spins interact. 
 For instance, if J < 0, it is clear that 
configurations with nearby spins ‘alig-
ned’ have lower energy, and hence 
higher probability. The relative weight of 
a configuration is 
 exp (–EV/T), 
where T is the temperature. The Ising 
model seeks to describe magnetic phe-
nomena, e.g. when J < 0, the above is a 
model for ferromagnetism. 
 The infinite-volume limit need not be 
unique, and it can vary non-analytically 
with the thermodynamic parameters. 
Points of non-uniqueness or non-analyti-
city in the parameter space are called 
phase transitions. In the case of the Ising 
model, one wishes to study how the 
magnetization of the infinite-volume 
limit (the model for a real magnet) varies 
with temperature T. As T is raised, mag-
netization weakens until, at a certain 
critical temperature Tc, it becomes zero. 
This is in fact a phase transition. One can 
say even more: in the Ising model, as 
well as in a large menagerie of models, 
correlations – which we shall not define 
rigorously (in the Ising model, ‘two-point 
correlation’ is the moment 〈σvσv′〉) –  
fall off with a power law (i.e. polyno-
mially as opposed to, say, exponentially) 
in terms of distance between the vertices 
v and v′. A consequence of this slow de-
cay of correlations at criticality is the 
following. By taking a suitable limit (in 
the case of the square lattice Γa, for ex-
ample, letting a → 0+, while simultane-
ously letting T → Tc), one is able to: 
 
• define thermodynamically meaningful 
measures that live on a space of paths 
in R2 (paths of a certain description that 
is dictated by the specific model), and 
• rigorously deduce information about 
the infinite-volume limit. 
 
This limit is called a scaling limit. Con-
sider again the Ising model: given a con-
figuration of spins, one can define a set 
of ‘contours’ – closed loops in the graph 
Γa – that separate regions of ‘up’ spins 
from regions of ‘down’ spins. The 
aforementioned ‘space of paths in R2 of a 
certain description’ in this case is the set 
of limits of all these contours. Indeed, 
this sort of construction is possible for a 
large zoo of other lattice models at criti-
cality. 
 An important postulate in the study of 
lattice models is the following. 
 
 
 
Figure 1. The square lattice Γa, lattice 
spacing a. 
PERSONAL NEWS 
 
CURRENT SCIENCE, VOL. 96, NO. 2, 25 JANUARY 2009 298 
Postulate/conjecture. Scaling limits of 
two-dimensional models at criticality are 
conformally invariant. 
 Given a lattice model and a region D 
in R2, let μD denote the aforementioned 
measure on the relevant ‘space of paths’ 
in D associated to the scaling limit. The 
above conjecture says that if D and D′ 
are two regions related by a conformal 
map Φ : D → D′, then the associated 
measures μD and μD ′ are related in terms 
of Φ. Much of the work by physicists in 
the past three decades assumes the exi-
stence of suitable scaling limits, as well 
as the conformal invariance of this limit. 
Under this assumption, techniques from 
representation theory are powerful enough to 
enable detailed computations to be made. 
 Schramm’s key contribution to all this 
originates from a study of the scaling 
limits of a lattice model known as the 
loop-erased random walk (LERW). The 
technology sketched out in the previous 
paragraphs applies to the LERW – any 
scaling limit of a LERW is supported on 
loop-free curves in the plane. The crux of 
Schramm’s analysis of the conformal-
invariance question (which forms a part 
of the influential article in Israel J. 
Math., 2000, 118, 221–288) lies in his 
discovery of a new family of manifestly 
conformally invariant stochastic processes 
which subsumes within it the scaling 
limit of (among other two-dimensional 
lattice models) the LERW. This stochas-
tic process is the SLE. In the last couple 
of years, the SLE has come to be identi-
fied with Schramm by name; it is now 
called the Schramm–Loewner evolution. 
 Here is a brief but lucid explanation of 
this beautiful and important idea, ex-
cerpted from the Laudatio delivered by 
Michael Aizenman on the occasion of the 
presentation (in 2003) of the Henri Poin-
caré Prize to Schramm. 
 “Let γ : [0, ∞) → D be a curve forming 
a slit of the open unit disk D, growing 
from a boundary point into the interior. 
For each t ≥ 0, let gt : D\γ [0, t] → D, be 
the Riemann map of the partly slit disk 
onto D, which is made unique by impos-
ing the conditions: gt(0) = 0, and 
gt′(0) > 0 is real and positive. As the slit 
is ‘unzipped’ the point of growth ξ (t) = 
gt(γ (t)) is moving along the boundary 
∂D. C. Loewner presented a procedure 
for solving the inverse problem: of re-
covering the curve γ (t) from ξ (t) and the 
rate information given by gt′(0) . . . Oded 
Schramm . . . applied Loewner’s prescrip-
tion in considering random curves with 
the property that the conditional distribu-
tion of a future segment of γ, conditioned 
on the past, depends in a conformally in-
variant way on the past trajectory. He no-
ticed that such a ‘conformal Markov 
property’ requires ξ (t) to evolve through 
independent increments, and thus to form 
a Brownian motion on the unit circle. . . 
This . . . led him to formulate the one-
parameter family of random paths 
SLEκ . . .” 
 To recapitulate: Schramm unveiled a 
new family of stochastic processes de-
scribed by the stochastic analogue of C. 
Loewner’s differential equation (cf. 
Math. Ann., 1923, 89, 103–121) and 
whose solutions must satisfy a Markov-
type condition that leads to conformal 
invariance. Schramm noticed that this 
‘conformal Markov property’ requires 
that the trajectory of ξ (t) describe a 
Brownian motion with some fixed diffu-
sion rate κ. For a given choice of κ, the 
stochastic process driven by the 
Brownian motion {ξ (t)|t ≥ 0} is called 
SLEκ. To complete the story of the 
LERW, the conformal invariance conjec-
ture for the LERW was settled by view-
ing the trajectory of a LERW in the open 
unit disc D as a slit that ‘unzips’ D, and 
proving that the scaling limit of the 
LERW is the stochastic process SLE2. 
 There is more to this story. Briefly: the 
stochastic processes SLEκ are natural 
candidates for the scaling limits of many 
two-dimensional lattice models at criti-
cality. In a series of deep papers, done 
mostly in collaboration with Greg Lawler 
and Wendelin Werner, Schramm establi-
shed the conformal invariance of the scaling 
limits of several two-dimensional lattice 
models, and was able to rigorously establish 
the conjectured values of certain critical 
exponents for these models. A sampling: 
1. They studied the scaling limit of per-
colation on a hexagonal lattice (con-
formal invariance had been proved 
earlier by Stanislav Smirnov) – 
specifically: a complete derivation of 
the critical exponents. 
2. In 1982, Mandelbrot conjectured that 
the fractal dimension of the boundary 
of the trajectory of a Brownian path is 
4/3. They proved this by establishing 
a connection with percolation. 
3. They rigorously derived the values of 
the ‘intersection exponents’ for Brow-
nian motion that had been conjectured 
by physicists Duplantier and Kwon (cf. 
Phys. Rev. Lett., 1988, 61, 2514–2517). 
A phenomenon and an inspiration 
Oded Schramm was widely considered to 
be the most influential probabilist of this 
decade. The awards that he received in-
clude the Anna and Lajos Erdös Prize in 
Mathematics in 1996, the Salem Prize in 
2001, the Clay Research Award in 2002, 
the Poincaré Prize in 2003 and the Pólya 
Prize in 2006. In 2008, he was elected as 
Member of the Royal Swedish Academy 
of Sciences. Wendelin Werner, whose 
Fields Medal of 2006 was the first 
awarded to a probabilist, recognized his 
debt to his collaborators Lawler and 
Schramm – and quoted, “the prize is also 
theirs, even if I am the only one under 
the age limit.” 
 Amidst all this international attention, 
Schramm remained a calm and humble 
individual. Everyone who knew him 
speaks of the loss of not just a fine 
mathematician, but of a very kind and 
understanding person – always patient, 
and generous in sharing his insights. 
GAUTAM BHARALI1,*  
T. R. RAMADAS2 
 
1Department of Mathematics,  
Indian Institute of Science,  
Bangalore 560 012, India 
2Abdus Salam International Centre for  
 Theoretical Physics,  
11 Strada Costiera,  
Trieste 34014, Italy 
*e-mail: bharali@math.iisc.ernet.in 
 
 
Figure 2. Loewner evolution showing a conformal transformation g := gT, T > t. 


Oded Schramm and the Schramm-Loewner evolution: in memoriam

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