Institute for applied mathematics

# Topics in Functional Integration and Multiscale Analysis, SS 2014

 Time and Room Monday and Wednesday, 12-14, Math/N 0.007 Neubau Start 7. April 2014

### Content

A large number of problems in modern statistical mechanics and theoretical physics can be translated into the study of suitable functional integrals. These are integrals over many variables, or more generally measures on spaces of functions or distributions.These are integrals over many variables, or more generally measures

on spaces of functions or distributions. They often take the form

$$\int_{\Omega}F(\varphi)d\mu(\varphi)$$, with  $$d\mu(\varphi) \propto \exp[-S(\varphi)]d\varphi$$,

where $$\Omega$$ is some space of functions and d$$\varphi$$ is the Lebesgue measure on $$\Omega$$.
Most of these integrals cannot be computed explicitely. Nevertheless much useful, precise information can be gained using relatively elementary, but mathematically rigorous, methods.

The key idea is to break the integral into "pieces", that are "small" enough to simplify their analysis, but "large" enough to contain the information we are looking for. The challenge is then to control the corresponding combinatorics. According to the problem at hand and the kind of information we need, there are different expansion schemes, among them cluster and Mayer expansions, renormalization, transfer matrix..

This course is to provide an introduction to some of these methods, as well as some applications. SInce a large class of problems both in statistical mechanics and in theoretical physics can be formulated in terms of perturbations of gaussian measures, therefore a large portion of the course will be dedicated to gaussian measures.

To avoid unnecessary additional technical complications, we will concentrate on the analysis of lattice functional integrals, as these give already rise to highly non trivial problems.

## Program and lecture notes

Chapter 1. Introduction: some motivations and examples (Gibbs measures in statistical mechanics)

Chapter 2. Problems in one dimension or in strips: transfer matrix technique.

• Ising model: φ = ±1 [Kup, Ch.1]

• Harmonic Kac operator and generalizations [Hel, Ch.5]

Chapter 3. Problems in general dimension.

• general facts on Gaussian measures  (the harmonic cristal revised)

• symmetries and complex deformations: low temperature correlations for the O(n) model

• duality transformations and saddle analysis: phase transition in the mean field case

### Some bibliography

[Hel] Bernard Helffer.

Semiclassical analysis, Witten Laplacians, and statistical mechanics,

volume 1 of Series in Partial Differential Equations and Applications. World Scientific Publishing Co., Inc., River Edge, NJ, 2002.

[Kup] Antti Kupiainen.

[MS] Oliver A. McBryan and Thomas Spencer.

On the decay of correlations in SO(n)-symmetric ferromagnets.

Comm. Math. Phys., 53(3):299–302, 1977.

## News

Prof. Jens Frehse (em.) hat die Golden Commemorative Medal der Fakultät für Mathematik und Physik der Karls-Universität Prag erhalten für sein außerordentliches wissenschaftliches Lebenswerk sowie für seine bedeutenden Beiträge zur Analysis Partieller Differentialgleichungen. (01.06.2018)

Die ehemalige Doktorandin am IAM, Lisa Hartung (aktuell: Courant Institute of Mathematical Sciences, New York), wurde für ihre herausragende Doktorarbeit "Extremal Processes in Branching Brownian Motion and Friends" (Betreuer: Prof. Bovier) mit dem Förderpreis der DMV-Fachgruppe Stochastik ausgezeichnet. Details hier. (14.03.2018)

Prof.  Patrik Ferrari erhält den ersten Alexandros Award zusammen mit Ivan Corwin (Columbia University) und Alexei Borodin (MIT) für den Artikel "Free energy fluctuations for directed polymers in random media in 1+1 dimensions".
Details siehe hier. (12.03.2018)

Frau Dr. Martina Vera Baar erhält den Hausdorff-Gedächtnispreis für die beste Dissertation des akademischen Jahres 2016/17 (Betreuer: Prof. Dr. A. Bovier, IAM). (Pressemitteilung, 24.01.2018)

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Managing Director: Prof. Dr. Anton Bovier