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Oded Schramm and the Schramm–Loewner evolution: In memoriam

Oded Schramm, the inventor of the sto- as limits of measures on suitably trun- phase transitions. In the case of the Ising chastic Loewner evolution (SLE), died cated finite-dimensional systems. Typi- model, one wishes to study how the on 1 September 2008 in a tragic hiking cally, there are only finitely many magnetization of the infinite-volume accident in the Cascades Mountains. He variables (dimensions) associated with a limit (the model for a real magnet) varies was forty-six. He is survived by his wife finite volume V – and the limit involves with temperature T. As T is raised, mag- and two children. taking V → ∞ (‘infinite volume’) limit. netization weakens until, at a certain Schramm left behind an impressive The approximating measures depend on critical temperature Tc, it becomes zero. legacy of research in probability and sto- the interaction energy EV of the specific This is in fact a phase transition. One can chastic processes, , system, as well as on thermodynamic say even more: in the Ising model, as and discrete and computational geo- parameters (e.g. temperature T, magnetic well as in a large menagerie of models, metry. He brought highly original mathe- field, etc.). correlations – which we shall not define matical insights to the study of critical In two dimensions, the theory is parti- rigorously (in the Ising model, ‘two-point phenomena in lattice models in statistical cularly rich, and a variety of techniques – correlation’ is the moment 〈σvσv′〉) – physics. Schramm will be sorely missed probabilistic, algebraic and heuristic – fall off with a power law (i.e. polyno- by the mathematics and the statistical have been deployed to great effect, start- mially as opposed to, say, exponentially) physics communities. ing with Onsager’s solution of the Ising in terms of distance between the vertices Oded Schramm was born in model. We shall use this familiar model v and v′. A consequence of this slow de- in December 1961. At the Hebrew Uni- as a guide to some of the technical con- cay of correlations at criticality is the versity, he received a B Sc in mathemat- structs that underpin the work of following. By taking a suitable limit (in ics and computer science, and an M Sc in Schramm and his colleagues. To begin the case of the square lattice Γa, for ex- mathematics under the direction of Gil with, one considers a ‘square lattice’, ample, letting a → 0+, while simultane- 2 Kalai. His Ph D (1990) was from Prince- namely the infinite grid Γa in R . (The ously letting T → Tc), one is able to: ton – his supervisor: . parameter a determines how fine a grid After a postdoctoral stint at the Univer- we get.) To each vertex v of the grid, one • define thermodynamically meaningful sity of California, San Diego, Schramm associates a variable (‘spin’) σv taking measures that live on a space of paths moved to the Weizmann Institute in values 1 (denoting ‘up’ spin) or –1 (de- in R2 (paths of a certain description that 1992. In 1999, he joined the Theory noting ‘down’ spin). The ‘configuration is dictated by the specific model), and Group at Research in Red- space’ consists of all assignments of spin • rigorously deduce information about mond. at every vertex. (One can view this space the infinite-volume limit. Schramm was an avid hiker. He listed as an infinite cartesian product of his interests – on his Microsoft webpage {1, −1} indexed by the vertices v.) Given This limit is called a scaling limit. Con- some years back – thus: ‘Percolation, a finite region V, the function EV is de- sider again the Ising model: given a con- two-dimensional random systems, criti- fined on the space ∏v∈V{1, −1} by the figuration of spins, one can define a set cal systems, SLE, conformal mappings, following formula: of ‘contours’ – closed loops in the graph dynamical random systems, discrete and Γa – that separate regions of ‘up’ spins EJ= σσ , coarse geometry, mountains’. Vvlvl∑ 12() () from regions of ‘down’ spins. The lV∈ aforementioned ‘space of paths in R2 of a where the sum runs over edges l of the certain description’ in this case is the set Conformal invariance of scaling grid contained in V, and vi(l) are the ver- of limits of all these contours. Indeed, limits and the SLE tices at either end of the edge l. Here J is this sort of construction is possible for a a real parameter that measures how large zoo of other lattice models at criti- The most celebrated of Schramm’s con- nearby spins interact. cality. tributions is his work on scaling limits of For instance, if J < 0, it is clear that An important postulate in the study of two-dimensional lattice models in statis- configurations with nearby spins ‘alig- lattice models is the following. tical mechanics. The goal of statistical ned’ have lower energy, and hence mechanics, roughly speaking, is to study higher probability. The relative weight of the thermodynamic properties of large a configuration is ensembles of interacting particles. The exp (–E /T), dimension of the ‘configuration space’ V associated with this ensemble is the where T is the temperature. The Ising number of its degrees of freedom. In the model seeks to describe magnetic phe- large-number limit, the configuration nomena, e.g. when J < 0, the above is a space is infinite-dimensional. In mathe- model for ferromagnetism. matical terms, statistical mechanics is The infinite-volume limit need not be concerned with the construction of thermo- unique, and it can vary non-analytically dynamically significant probability with the thermodynamic parameters. measures on these infinite dimensional Points of non-uniqueness or non-analyti- Figure 1. The square lattice Γa, lattice spaces. These measures are constructed city in the parameter space are called spacing a.

CURRENT SCIENCE, VOL. 96, NO. 2, 25 JANUARY 2009 297 PERSONAL NEWS

mostly in collaboration with and , 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 Figure 2. Loewner evolution showing a conformal transformation g := gT, T > t. earlier by ) – specifically: a complete derivation of Postulate/conjecture. Scaling limits of the Riemann map of the partly slit disk the critical exponents. two-dimensional models at criticality are onto D, which is made unique by impos- 2. In 1982, Mandelbrot conjectured that conformally invariant. ing the conditions: gt(0) = 0, and the fractal dimension of the boundary Given a lattice model and a region D gt′(0) > 0 is real and positive. As the slit of the trajectory of a Brownian path is in R2, let denote the aforementioned is ‘unzipped’ the point of growth (t) = μD ξ 4/3. They proved this by establishing measure on the relevant ‘space of paths’ gt(γ (t)) is moving along the boundary a connection with percolation. in D associated to the scaling limit. The ∂D. C. Loewner presented a procedure 3. They rigorously derived the values of above conjecture says that if D and D′ for solving the inverse problem: of re- the ‘intersection exponents’ for Brow- are two regions related by a conformal covering the curve γ (t) from ξ (t) and the nian motion that had been conjectured map Φ : D → D′, then the associated rate information given by gt′(0) . . . Oded by physicists Duplantier and Kwon (cf. measures and are related in terms Schramm . . . applied Loewner’s prescrip- μD μD ′ Phys. Rev. Lett., 1988, 61, 2514–2517). of Φ. Much of the work by physicists in tion in considering random curves with the past three decades assumes the exi- the property that the conditional distribu- A phenomenon and an inspiration stence of suitable scaling limits, as well tion of a future segment of γ, conditioned as the conformal invariance of this limit. on the past, depends in a conformally in- Oded Schramm was widely considered to Under this assumption, techniques from variant way on the past trajectory. He no- be the most influential probabilist of this representation theory are powerful enough to ticed that such a ‘conformal Markov decade. The awards that he received in- enable detailed computations to be made. property’ requires ξ (t) to evolve through clude the Anna and Lajos Erdös Prize in Schramm’s key contribution to all this independent increments, and thus to form Mathematics in 1996, the Salem Prize in originates from a study of the scaling a Brownian motion on the unit circle. . . 2001, the in 2002, limits of a lattice model known as the This . . . led him to formulate the one- the Poincaré Prize in 2003 and the Pólya loop-erased (LERW). The parameter family of random paths Prize in 2006. In 2008, he was elected as technology sketched out in the previous SLEκ . . .” Member of the Royal Swedish Academy paragraphs applies to the LERW – any To recapitulate: Schramm unveiled a of Sciences. Wendelin Werner, whose scaling limit of a LERW is supported on new family of stochastic processes de- of 2006 was the first loop-free curves in the plane. The crux of scribed by the stochastic analogue of C. awarded to a probabilist, recognized his Schramm’s analysis of the conformal- Loewner’s differential equation (cf. debt to his collaborators Lawler and invariance question (which forms a part Math. Ann., 1923, 89, 103–121) and Schramm – and quoted, “the prize is also of the influential article in J. whose solutions must satisfy a Markov- theirs, even if I am the only one under Math., 2000, 118, 221–288) lies in his type condition that leads to conformal the age limit.” discovery of a new family of manifestly invariance. Schramm noticed that this Amidst all this international attention, conformally invariant stochastic processes ‘conformal Markov property’ requires Schramm remained a calm and humble which subsumes within it the scaling that the trajectory of ξ (t) describe a individual. Everyone who knew him limit of (among other two-dimensional Brownian motion with some fixed diffu- speaks of the loss of not just a fine lattice models) the LERW. This stochas- sion rate κ. For a given choice of κ, the mathematician, but of a very kind and tic process is the SLE. In the last couple stochastic process driven by the understanding person – always patient, of years, the SLE has come to be identi- Brownian motion {ξ (t)|t ≥ 0} is called and generous in sharing his insights. fied with Schramm by name; it is now SLEκ. To complete the story of the 1, called the Schramm–Loewner evolution. LERW, the conformal invariance conjec- GAUTAM BHARALI * 2 Here is a brief but lucid explanation of ture for the LERW was settled by view- T. R. RAMADAS this beautiful and important idea, ex- ing the trajectory of a LERW in the open cerpted from the Laudatio delivered by unit disc D as a slit that ‘unzips’ D, and 1Department of Mathematics, Michael Aizenman on the occasion of the proving that the scaling limit of the Indian Institute of Science, presentation (in 2003) of the Henri Poin- LERW is the stochastic process SLE2. Bangalore 560 012, India caré Prize to Schramm. There is more to this story. Briefly: the 2Abdus Salam International Centre for

“Let γ : [0, ∞) → D be a curve forming stochastic processes SLEκ are natural Theoretical Physics, a slit of the open unit disk D, growing candidates for the scaling limits of many 11 Strada Costiera, from a boundary point into the interior. two-dimensional lattice models at criti- Trieste 34014, Italy

For each t ≥ 0, let gt : D\γ [0, t] → D, be cality. In a series of deep papers, done *e-mail: [email protected]

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