About the Book
This volume is dedicated to the memory of the late Oded Schramm (1961-2008), distinguished mathematician. Throughout his career, Schramm made profound and beautiful contributions to mathematics that will have a lasting influence.
In these two volumes, Editors Itai Benjamini and Olle Häggström have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Häggström and Cristophe Garban that further elucidate his work. The papers within are a representative collection that shows the breadth, depth, enthusiasm and clarity of his work, with sections on Geometry, Noise Sensitivity, Random Walks and Graph Limits, Percolation, and finally Schramm-Loewner Evolution. An introduction by the Editors and a comprehensive bibliography of Schramm's publications complete the volume. The book will be of especial interest to researchers in probability and geometry, and in the history of these subjects.
Table of Contents:
Volume 1.- Part 1: Geometry.- Commentary: Oded Schramm: From Circle Packing to SLE, by Steffen Rohde. To appear in Ann. Probab. Reprinted with permission of Institute of Mathematical Statistics.- O. Schramm. Illuminating sets of constant width. Mathematika Vol. 35 No. 2, 180--189 (1988). Reprinted with permission of Cambridge University Press.- O. Schramm. On the volume of sets having constant width. Israel J. Math. Vol. 63 No. 2, 178--182 (1988). Reprinted with permission of Hebrew University Magnes Press.- O. Schramm. Rigidity of infinite (circle) packings. J. Amer. Math. Soc. Vol. 4 No. 1, 127--149 (1991). Reprinted with permission of American Mathematical Society.- O. Schramm. How to cage an egg. Invent. Math. Vol. 107 No. 3, 543--560 (1992). Reprinted with permission of Springer Science+Business Media.- Zheng-Xu He and O. Schramm. Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2) Vol. 137 No. 2, 369--406 (1993). Reprinted with permission of Princeton University and the Institute for Advanced Study.- Zheng-Xu He and O. Schramm. Hyperbolic and parabolic packings. Discrete Comput. Geom. Vol. 14 No. 2, 123--149 (1995). Reprinted with permission of Springer Science+Business Media.- O. Schramm. Circle patterns with the combinatorics of the square grid. Duke Math. J. Vol. 86 No. 2, 347--389 (1997). Reprinted with permission of Duke University Press.- Zheng-Xu He and O. Schramm. The C∞-convergence of hexagonal disk packings to the Riemann map. Acta Math. Vol. 180 No. 2, 219--245 (1998). Reprinted with permission of Springer Science+Business Media.- M. Bonk and O. Schramm. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. Vol. 10 No. 2, 266--306 (2000). Reprinted with permission of Springer Science+Business Media.- Part 2: Noise Sensitivity.- Commentary: Oded Schramm’s Contributions to Noise Sensitivity, by Christophe Garban. To appear in Ann.Probab. Reprinted with permission of Institute of Mathematical Studies.- I. Benjamini, G. Kalai, and O. Schramm. Noise sensitivity of boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. No. 90, 5--43 (1999). Reprinted with permission of Institut Des Hautes Études Scientifiques.- O. Schramm and J. E. Steif. Quantitative noise sensitivity and exceptional times for percolation. Annals of Mathematics, Pages 619-672 from Volume 171 (2010), Issue 2. Reprinted with permission of Princeton University and the Institute for Advanced Study.- C. Garban, G. Pete, and O. Schramm. The Fourier Spectrum of Critical Percolation. Acta Math. 205 (2010), 19-104. Reprinted with permission of Springer Science+Business Media.- Part 3: Random Walks and Graph Limits.- I. Benjamini and O. Schramm. Recurrence of Distributional Limits of Finite Planar Graphs. Electron. J. Probab. Vol. 6, no. 23, 13 pp. (electronic) (2001). Reprinted with permission of Institute of Mathematical Statistics.- O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys. Vol. 241 No. 2-3, 191--213 (2003). Reprinted with permission of Springer Science+Business Media.- O. Schramm. Compositions of random transpositions. Israel J. Math. Vol. 147, 221--243 (2005). Reprinted with permission of Hebrew University Magnes Press.- Y. Peres, O.Schramm, S.Sheffield, and D.B. Wilson. Tug-of-war and the infinity Laplacian. Journal of the American Mathematical Society 22(1):167--210, (2009). Reprinted with permission of Yuval Peres.- O. Schramm. Hyperfinite graph limits. Electron. Res. Announc. Math. Sci. Vol. 15, 17--23 (2008). Reprinted with permission of the American Institute of Mathematical Sciences.- Volume 2.- Part 4: Percolation.- Commentary: Percolation beyond Z^d: The Contributions of Oded Schramm, by Olle Häggström. To appear in Ann. Probab. Reprinted with permission of Institute ofMathematical Statistics.- I. Benjamini and O. Schramm. Percolation beyond Zd, many questions and a few answers. Electron. Comm. Probab. Vol. 1, no.\ 8, 71--82 (electronic) (1996). Reprinted with permission of Institute of Mathematical Statistics.- I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. Vol. 27 No. 3, 1347--1356 (1999). Reprinted with permission of Institute of Mathematical Statistics.- R. Lyons and O. Schramm. Indistinguishability of Percolation Clusters. Ann. Probab. Vol. 27 No. 4, 1809--1836 (1999). Reprinted with permission of Institute of Mathematical Statistics.- I. Benjamini and O. Schramm. Percolation in the Hyperbolic Plane. J. Amer. Math. Soc. Vol. 14 No. 2, 487--507 (electronic) (2001). Reprinted with permission of American Mathematical Society.- I. Benjamini, H. Kesten, Y. Peres, and O. Schramm. Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12, ... Ann. of Math. (2) Vol. 160 No. 2, 465--491 (2004). Reprinted with permission of Princeton University and the Institute for Advanced Study.- I. Benjamini, G. Kalai, and O.Schramm. First passage percolation has sublinear distance variance. Ann. Probab. Vol. 31 No. 4, 1970--1978 (2003). Reprinted with permission of Institute of Mathematical Statistics.- Part 5: Schramm-Loewner Evolution.- O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. Vol. 118, 221--288 (2000). Reprinted with permission of Hebrew University Magnes Press.- G.F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. Vol. 187 No. 2, 237--273 (2001). Reprinted with permission of Springer Science+Business Media.- G.F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents II: Plane exponents. Acta Math. Vol. 187 No. 2,275--308 (2001). Reprinted with permission of Springer Science+Business Media.- G.F. Lawler, O. Schramm, and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. Vol. 32 No. 1B, 939--995 (2004). Reprinted with permission of Institute of Mathematical Statistics.- S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) Vol. 161 No. 2, 883--924 (2005). Reprinted with permission of Princeton University and the Institute for Advanced Study.- O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Mathematica, Volume 202, Number 1, 21-137, (2009). Reprinted with permission of Springer Science+Business Media.- O. Schramm, S. Sheffield, and D. B. Wilson. Conformal radii for conformal loop ensembles. Commun.Math.Phys.288: 43-53, (2009). Reprinted with permission of Springer Science+Business Media.- O. Schramm. Conformally invariant scaling limits: an overview and a collection of problems. In International Congress of Mathematicians. Vol. I, 513--543, Eur. Math. Soc., Zürich (2007). Reprinted with permission of International Mathematical Union.- O. Schramm and S. Smirnov. On the scaling limits of planar percolation. To appear in Ann. Probab. Reprinted with permission of Institute of Mathematical Statistics.
