We see how the underling geometric properties of
space, usually a graph manifest themselves in the qualitative behavior
of random processes, in particular random walk and percolation, taking
place on the graph. For example geometric conditions for Liouville
theorems and uniqueness of the infinite percolation cluster.

**References**

- Woess, Wolfgang Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp.
- Grimmett, Geoffrey Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp.
- Grimmett, Geoffrey Percolation and disordered systems. Lectures on
probability theory and statistics (Saint-Flour, 1996), 153--300, Lecture
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**Papers**: - Benjamini, I.; Schramm, O. Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. 7 (1997), no. 3, 403--419.
- math.PR/0306355 A Phase Transition for the Metric Distortion of Percolation on the Hypercube. Omer Angel, Itai Benjamini. 11 pages.
- math.MG/0303127 Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality. Itai Benjamini, Oded Schramm. 5 pages
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- T. W. Anderson. The integral of a symmetric function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 (1955) 170-176
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Mathematics 1617. Springer-Verlag 1995

**K.Johansson**,*Determinantal Processes in Random Matrix Theory*The eigenvalues of certain ensembles of Hermitian matrices are determinantal point processes. Similar point processes occur naturally also in other contexts, like random permutations, random growth and random tilings. Often these processes can be constructed using non-interesecting paths. The fact that we get determinantal processes in these other problems leads to similarities between these problems and random matrix theory.

- P. Forrester, "Log-gases and random matrices", book manuscript available at http://www.ms.unimelb.edu.au/~matpjf/matpjf.html
- M. L. Mehta, Random matrices.
- K. Johansson, Toeplitz determinants, random growth and determinantal processes. Kurt Johansson. Proceedings of the ICM, Beijing 2002, vol. 3, 53--62, math.PR/0304368.
- C.A. Tracy, H. Widom, Distribution Functions for Largest Eigenvalues and Their Applications. In Proceedings of the International Congress of Mathematicians, Beijing 2002, Vol. I, ed. LI Tatsien, Higher Education Press, Beijing 2002, pgs. 587-596.

**G.Lugosi**,*Concentration of Functions of Independent Random Variables*The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Interestingly, such a concentration behavior is shared by a large class of general functions of independent random variables. The purpose of this course is to give an introduction to some of these general concentration inequalities.

The concentration-of-measure phenomenon has been known and widely used since the 1970's in various fields of applications but in the last decade it has received renewed attention thanks to various breakthrough results of Talagrand.

In this course simple conditions will be derived under which such concentration inequalities may be established. Several applications will be shown in various areas including empirical process theory, random combinatorics, and statistical learning theory.

**References**- J.Baik, P.Deift, and K.Johansson. On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geometric and Functional Analysis, 10:702--731, 2000.
- W.Beckner. A generalized Poincar\'e inequality for Gaussian measures. Proceedings of the American Mathematical Society, 105:397--400, 1989.
- S.Bobkov and M.Ledoux. Poincar\'e's inequalities and Talagrands's concentration phenomenon for the exponential distribution. Probability Theory and Related Fields, 107:383--400, 1997.
- S.Boucheron, G.Lugosi, and P.Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.
- S.Boucheron, G.Lugosi, and P.Massart. Concentration inequalities using the entropy method. Annals of Probability, to appear, 2003.
- O.Bousquet. A Bennett concentration inequality and its application to suprema of empirical processes. C. R. Acad. Sci. Paris, 334:495--500, 2002.
- O.Bousquet. New approaches to statistical learning theory. Annals of the Institute of Statistical Mathematics, 2003.
- D.Chafai. On $\phi$-entropies and $\phi$-Sobolev inequalities. Technical report, arXiv.math.PR/0211103, 2002.
- A.Dembo. Information inequalities and concentration of measure. Annals of Probability, 25:927--939, 1997.
- L.Devroye. Exponential inequalities in nonparametric estimation. In G.Roussas, editor, Nonparametric Functional Estimation and Related Topics, pages 31--44. NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 1991.
- B.Efron and C.Stein. The jackknife estimate of variance. Annals of Statistics, 9:586--596, 1981.
- S.Janson, T.Luczak, and A.Ruci\'nski. Random graphs. John Wiley, New York, 2000.
- R.Latala and C.Oleszkiewicz. Between Sobolev and Poincar\'e. In Geometric Aspects of Functional Analysis, Israel Seminar (GAFA), 1996-2000, pages 147--168. Springer, 2000. Lecture Notes in Mathematics, 1745.
- M.Ledoux. Isoperimetry and gaussian analysis. In P.Bernard, editor, Lectures on Probability Theory and Statistics, pages 165--294. Ecole d'Eté de Probabilités de St-Flour XXIV-1994, 1996.
- M.Ledoux. On Talagrand's deviation inequalities for product measures. ESAIM: Probability and Statistics, 1:63--87, 1997. \tt http://www.emath.fr/ps/.
- M.Ledoux. Concentration of measure and logarithmic sobolev inequalities. In Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics 1709, pages 120--216. Springer, 1999.
- M.Ledoux. The concentration of measure phenomenon. American Mathematical Society, Providence, RI, 2001.
- Malwina J. Luczak and Colin McDiarmid. Concentration for locally acting permutations. Discrete Mathematics, page to appear, 2003.
- K.Marton. A simple proof of the blowing-up lemma. IEEE Transactions on Information Theory, 32:445--446, 1986.
- K.Marton. Bounding $\bar{d}$-distance by informational divergence: a way to prove measure concentration. Annals of Probability, 24:857--866, 1996.
- K.Marton. A measure concentration inequality for contracting Markov chains. Geometric and Functional Analysis, 6:556--571, 1996. Erratum: 7:609--613, 1997.
- P.Massart. About the constants in Talagrand's concentration inequalities for empirical processes. Annals of Probability, 28:863--884, 2000.
- P.Massart. Some applications of concentration inequalities to statistics. Annales de la Facult\'e des Sciencies de Toulouse, IX:245--303, 2000.
- C.McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989, pages 148--188. Cambridge University Press, Cambridge, 1989.
- C.McDiarmid. Concentration. In M.Habib, C.McDiarmid, J.Ramirez-Alfonsin, and B.Reed, editors, Probabilistic Methods for Algorithmic Discrete Mathematics, pages 195--248. Springer, New York, 1998.
- C.McDiarmid. Concentration for independent permutations. Combinatorics, Probability, and Computing, 2:163--178, 2002.
- D.Panchenko. A note on Talagrand's concentration inequality. Electronic Communications in Probability, 6, 2001.
- D.Panchenko. Symmetrization approach to concentration inequalities for empirical processes. Annals of Probability, to appear, 2003.
- W.Rhee and M.Talagrand. Martingales, inequalities, and NP-complete problems. Mathematics of Operations Research, 12:177--181, 1987.
- E.Rio. In\'egalit\'es de concentracion pour les processus empiriques de classes de parties. Probability Theory and Related fields, 119:163--175, 2001.
- P.-M. Samson. Concentration of measure inequalities for Markov chains and $\phi$-mixing processes. Annals of Probability, 28:416--461, 2000.
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- M.Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Publications Math\'ematiques de l'I.H.E.S., 81:73--205, 1995.
- M.Talagrand. New concentration inequalities in product spaces. Inventiones Mathematicae, 126:505--563, 1996.
- M.Talagrand. A new look at independence. Annals of Probabilit, 24:1--34, 1996. (Special Invited Paper).

**R. Schneider**,*Convexity in Stochastic Geometry*Stochastic geometry studies randomly generated geometric objects. We will give an introduction to two important models from stochastic geometry, which are based on stochastic processes of convex particles. In both of them, Poisson processes play a dominant role. The first example are the Boolean models, random closed sets generated as the union sets of Poisson processes of convex bodies. We prove a fundamental result about relations between functional densities of the union set and of the underlying particle process. The second example are random mosaics, in particular those induced from Poisson processes in the space of hyperplanes or by the Dirichlet-Voronoi cells of Poisson point processes. Here we will mainly be interested in asymptotic shapes of large cells.

**References**- Schneider, R. and Weil, W.: Stochastische Geometrie. Teubner, Stuttgart 2000.
- Schneider, R. and Weil, W.: Integralgeometrie. Teubner, Stuttgart 1992.
- Molchanov, I.S.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, New York 1997.
- Mřller, J.: Lectures on Random Voronoi Tessellations. Lect. Notes Statist. 87, Springer New York, 1994.
- Hug, D., Reitzner, M. and Schneider, R.: The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Prob. 32 (2004), 1140-1167.
- Hug, D., Reitzner, M. and Schneider, R.: Large Poisson-Voronoi cells and Crofton cells. Adv. Appl. Prob. 36 (2004), 667-690.