Ebook: Ten Lectures on Random Media

The following notes grew out oflectures held during the DMV-Seminar on Random Media in November 1999 at the Mathematics Research Institute of Oberwolfach, and in February-March 2000 at the Ecole Normale Superieure in Paris. In both places the atmosphere was very friendly and stimulating. The positive response of the audience was encouragement enough to write up these notes. I hope they will carryover the enjoyment of the live lectures. I whole heartedly wish to thank Profs. Matthias Kreck and Jean-Franc;ois Le Gall who were respon­ sible for these two very enjoyable visits, Laurent Miclo for his comments on an earlier version of these notes, and last but not least Erwin Bolthausen who was my accomplice during the DMV-Seminar. A Brief Introduction The main theme of this series of lectures are "Random motions in random me­ dia". The subject gathers a variety of probabilistic models often originated from physical sciences such as solid state physics, physical chemistry, oceanography, biophysics . . . , in which typically some diffusion mechanism takes place in an inho­ mogeneous medium. Randomness appears at two levels. It comes in the description of the motion of the particle diffusing in the medium, this is a rather traditional point of view for probability theory; but it also comes in the very description of the medium in which the diffusion takes place.

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The field of random media has been the object of intensive activity over the last twenty-five years. It gathers a variety of models generally originating from physical sciences, where certain materials or substances have defects or inhomogeneities. This feature can be taken into account by letting the medium be random. Randomness in the medium turns out to cause very unexpected effects, especially in the large-scale behavior of some of these models. What in the beginning was often deemed to be a simple toy-model ended up as a major mathematical challenge. After more than twenty years of intensive research in this field, certain new paradigms and some general methods have emerged, and the surprising results on the asymptotic behavior of individual models are now better understood in more general frameworks. This monograph grew out of the DMV Lectures on Random Media held by the authors at the Mathematical Research Institute in Oberwolfach in November 1999 and tries to give an account of some of the developments in the field, especially in the area of random motions in random media and of mean-field spin glasses. It will be a valuable resource for postgraduates and researchers in probability theory and mathematical physics.

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The field of random media has been the object of intensive activity over the last twenty-five years. It gathers a variety of models generally originating from physical sciences, where certain materials or substances have defects or inhomogeneities. This feature can be taken into account by letting the medium be random. Randomness in the medium turns out to cause very unexpected effects, especially in the large-scale behavior of some of these models. What in the beginning was often deemed to be a simple toy-model ended up as a major mathematical challenge. After more than twenty years of intensive research in this field, certain new paradigms and some general methods have emerged, and the surprising results on the asymptotic behavior of individual models are now better understood in more general frameworks. This monograph grew out of the DMV Lectures on Random Media held by the authors at the Mathematical Research Institute in Oberwolfach in November 1999 and tries to give an account of some of the developments in the field, especially in the area of random motions in random media and of mean-field spin glasses. It will be a valuable resource for postgraduates and researchers in probability theory and mathematical physics.
Content:
Front Matter....Pages i-vii
Front Matter....Pages 1-3
A Brief Introduction....Pages 5-7
The Environment Viewed from the Particle....Pages 9-15
Central Limit Theorem for Random Walks in Random Environment with Null Drift....Pages 16-22
Long Time Survival among Random Traps....Pages 23-31
Multi-Dimensional Random Walks in Random Environment....Pages 32-39
More on Random Walks in Random Environment....Pages 40-51
Front Matter....Pages 53-53
On the Sherrington-Kirkpatrick Model of Spin Glasses....Pages 55-63
The Sherrington-Kirkpatrick Model: High Temperature and Nonzero Magnetic Field....Pages 64-73
The Random Energy Model....Pages 74-82
The Generalized Random Energy Model and Induced Clusterings....Pages 83-95
Markovian Clustering, Reshuffling, and a Self-Consistency Equation....Pages 96-109
Back Matter....Pages 111-119

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The field of random media has been the object of intensive activity over the last twenty-five years. It gathers a variety of models generally originating from physical sciences, where certain materials or substances have defects or inhomogeneities. This feature can be taken into account by letting the medium be random. Randomness in the medium turns out to cause very unexpected effects, especially in the large-scale behavior of some of these models. What in the beginning was often deemed to be a simple toy-model ended up as a major mathematical challenge. After more than twenty years of intensive research in this field, certain new paradigms and some general methods have emerged, and the surprising results on the asymptotic behavior of individual models are now better understood in more general frameworks. This monograph grew out of the DMV Lectures on Random Media held by the authors at the Mathematical Research Institute in Oberwolfach in November 1999 and tries to give an account of some of the developments in the field, especially in the area of random motions in random media and of mean-field spin glasses. It will be a valuable resource for postgraduates and researchers in probability theory and mathematical physics.
Content:
Front Matter....Pages i-vii
Front Matter....Pages 1-3
A Brief Introduction....Pages 5-7
The Environment Viewed from the Particle....Pages 9-15
Central Limit Theorem for Random Walks in Random Environment with Null Drift....Pages 16-22
Long Time Survival among Random Traps....Pages 23-31
Multi-Dimensional Random Walks in Random Environment....Pages 32-39
More on Random Walks in Random Environment....Pages 40-51
Front Matter....Pages 53-53
On the Sherrington-Kirkpatrick Model of Spin Glasses....Pages 55-63
The Sherrington-Kirkpatrick Model: High Temperature and Nonzero Magnetic Field....Pages 64-73
The Random Energy Model....Pages 74-82
The Generalized Random Energy Model and Induced Clusterings....Pages 83-95
Markovian Clustering, Reshuffling, and a Self-Consistency Equation....Pages 96-109
Back Matter....Pages 111-119
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