BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:The polaron problem and Kac type interactions - Erwin Bolthausen (
 Zurich)
DTSTART:20260217T140000Z
DTEND:20260217T150000Z
UID:TALK244660@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:This is survey on some probabilistic problems motivated by the
 \nstrong coupling limit of the Fr\\"{o}hlich polaron (after H. Fr\\"{o}hli
 ch\n1952). This model describes the movement of a charged particle in a la
 ttice with polarized nodes. In physics\, the description is usually given 
 in terms of quantum mechanics and field theory. Feynman in his lectures on
 \nStatistical Mechanics 1972  gave in a purely probabilistic description o
 f\nsome of the key quantities. In particular he predicted that the effecti
 ve\nmass is proportional to $\\alpha^4$ where $\\alpha$ is the coupling\np
 arameter. The effective mass then equals the inverse of the variance\npara
 meter of a 3-dimensional Brownian motion with an interaction which is long
  range in time\, and given by an attractive Coulomb force in space. The mo
 del is in essence a delicate continuous time statistical model with a sing
 ular Kac type interaction. The latter means that there is a small Kac\npar
 ameter $\\lambda$\, and for $\\lambda \\rightarrow 0$\, the interaction\nb
 ecomes weak but long range in time\, so that it should become in the limit
  a mean-field type model. A key feature is that there is a continuous symm
 etry in space. In the polaron\, $\\lambda $ equals $\\alpha ^{-2}.$ Based 
 on this translation\, there is a short and very appealing\, but far from r
 igorous\, old argument by Spohn 1987 that predicts the correct asymptotic 
 behavior. The result itself has\, after many efforts\, recently been prove
 d by two groups: Bazaes\, Mukherjee\, Varadhan\, Sellke\, and Brooks\, Sei
 ringer\, the latter in the quantum mechanical setup. The first group uses 
 a delicate expansion of the Brownian interaction\, which bypasses the Kac-
 picture advocated by Spohn. The latter addresses a situation that is much 
 more general than the one appearing in the polaron\, and there still remai
 n interesting open problems on which there is some recent progress togethe
 r with Amir Dembo.
LOCATION:MR12
END:VEVENT
END:VCALENDAR