Ten Lectures on Random Media by Erwin Bolthausen

By Erwin Bolthausen

The following notes grew out oflectures held in the course of the DMV-Seminar on Random Media in November 1999 on the arithmetic examine Institute of Oberwolfach, and in February-March 2000 on the Ecole Normale Superieure in Paris. In either locations the ambience used to be very pleasant and stimulating. The confident reaction of the viewers used to be encouragement sufficient to jot down up those notes. i'm hoping they're going to carryover the joy of the reside lectures. I complete heartedly desire to thank Profs. Matthias Kreck and Jean-Franc;ois Le Gall who have been respon­ sible for those very relaxing visits, Laurent Miclo for his reviews on an prior model of those notes, and final yet now not least Erwin Bolthausen who was once my companion throughout the DMV-Seminar. a quick advent the most subject of this sequence of lectures are "Random motions in random me­ dia". the topic gathers numerous probabilistic versions frequently originated from actual sciences comparable to sturdy nation physics, actual chemistry, oceanography, biophysics . . . , during which generally a few diffusion mechanism happens in an inho­ mogeneous medium. Randomness appears to be like at degrees. It is available in the outline of the movement of the particle diffusing within the medium, this can be a particularly conventional standpoint for likelihood conception; however it additionally is available in the very description of the medium during which the diffusion takes place.

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1 of Saloff-Coste [58]). 27) is smaller than M( n ) sup iE[l,M(n)] 1P[ IOw n Bil < -C] . 22) follows. 11), we see that 10gS(n) '" 10gIE[(I- Aw(Tn))n]. The fact that log IE[(I-Aw(Tn))n] '" log IE [exp{ -nAw(Tn)}] is also straightforward. 22) shows that on the event {Aw(Tn) ~ 2 1 d+2 }, a "hole" appears in scale n d+2 in the trap configuration. One can go much further with the "method of enlargement of obstacles" and see that conditionally on the occurrence of an atypically low eigenvalue in a box, with high probability there is formation of a spherical hole, see for instance Chapter 4 §4 A in [64], and also Proposition 2 of Povel [51].

Somewhat in a similar spirit, traps provide a way to slowdown random walks in random environment. 3. 16)). Proof. 7) 1 = IBI I L IE[ xEBL L Px,w[TBL > n]] > IBLllE[exp{ -nAw{Bd}]· o The condition (T): We shall further explore the connection between traps and slowdowns of random walks in random environment. It will be convenient to this end to define a class of random walks in random environment which display a "ballistic behavior", when d ~ 2. Let us incidentally mention that questions concerning recurrence and transience of random walks in random environment are presently very poorly understood.

In U, and the infimum Ro(d, v) = (d:A~J di2 , (see for instance Chapter 4 §5 of [64]). 4. 32) there are no obstacle (d = 2), few obstacles (in the capacity sense, d:2: 3) 1 inside B(Dt, (Ro - €(t» t d +2 ) . The analogue when d = 1, can be found in Schmock [59]. 3 of [64] and the results of [51]. 2 by Bolthausen [7]. The case of dimension d :2: 3 remained open for quite some time. One of the reasons is that a simply connected domain of IRd can have a boundary with (d - I)-surface measure close to that of a ball with same volume, but still be far from any ball in the Hausdorff sense.

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