## Project 2: Systems with Many Degrees of Freedom: Probabilistic and Constructive Field Theory Methods

### Participants

Jean-Dominique Deuschel, Jürgen
Gärtner, **Wolfgang
König**, Manfred Salmhofer
### Summary

Building on our progress of the first funding period, we will consider certain types of complex systems with many degrees of freedom and investigate new fundamental aspects of their behaviour. For bosonic many-particle systems, we want to find mathematical evidence for Bose-Einstein condensation at fixed temperature, and we want to study the effect of an additional attraction of the particles at positive distance at vanishing temperature. For fermionic many-particle systems, we want to extend the analysis we achieved in the first funding period to two-dimensional systems and to three-dimensional systems under a weaker diluteness assumption. In the parabolic Anderson model we want to study aging properties of the system and correlations in time, in the hope to investigate random hopping dynamics in general. Again, all these tasks require a systematic combination of probabilistic and analytic tools.
### Some related earlier preprints

J. Gärtner and W. König:
**The parabolic Anderson model**,

in: J.-D. Deuschel and A. Greven (Eds.), *Interacting Stochastic Systems*,
pp. 153--179, Springer (2005).

R. van der Hofstad, W. König and
P. Mörters:
**The universality classes in the parabolic Anderson model**,

Commun. Math. Phys. **267:2**, 307-353 (2006).

Preprint
ps,
revised version,
pdf.

J.
Gärtner, W. König and S. Molchanov:
**Geometric characterization
of intermittency in the parabolic Anderson model,**

*Ann. Probab.* ** 35** (2007), 439-499.

Publication

J.
Gärtner, M. Heydenreich:
**Annealed asymptotics for the parabolic Anderson model with a moving catalyst,**

*Stoch. Proc. Appl.* **116** (2006), 1511-1529.

Publication

J.
Gärtner and F. den Hollander:
**Intermittency in a catalytic random medium,**

*Ann. Probab.* **34**, 2219-2287 (2006).

Preprint in Math ArXive.

## Achievements of the Research Group (funding period 2006/08)

Walter Pedra, Manfred Salmhofer:
**Determinant bounds and the Matsubara UV problem of many-fermion systems**

*Comm. Math. Phys.*, to appear (2008).

DOI 10.1007/s00220-008-0463-z
M. Birkner and R.~Sun :
**Annealed vs. quenched critical points for a random walk pinning model,**

preprint.

J.
Gärtner, F. den Hollander, and G. Maillard:
**
Intermittency on catalysts: symmetric exclusion,**

*Elec. Jour. Probab.* **12** (2007), paper no. 18, 516-573.

Publication

J. Gärtner and R. Sun:
**A quenched limit theorem for the local time of random walks on \Z^2,**

preprint.

J. Gärtner, F. den Hollander and G. Maillard:
**Intermittency on catalysts**

In: J. Blath, P. Mörters and M. Scheutzow (eds), Trends in Stochastic
Analysis.
LMS 353, Cambridge Univ. Press. In print (2008).

Preprint.

W. König, H. Lacoin, P. Mörters and N. Sidorova:
**A two cities theorem for the parabolic Anderson model,**

Preprint,
ps, Preprint,
pdf, *Ann. Probab.*, to appear.

G. Grüninger and W. König:
**Potential confinement property of the parabolic
Anderson model,**

Preprint,
ps, Preprint,
pdf.

B. Niethammer and J. Velasquez:
**On screening induced fluctuations in Ostwald ripening,**

Preprint, pdf, *J. Stat. Phys.* **130:3**, 415-453 (2008).

B. Niethammer and J. Velasquez:
**Screening in interacting particle systems,**

preprint, pdf (2008).

B. Niethammer:
**Effective theories for Ostwald ripening,**

preprint, pdf (2008).

S. Conti, A. Hönig, B. Niethammer, F. Otto:
**Non-universality in low-volume-fraction Ostwald ripening,**

preprint, pdf (2006).

M. Herrmann, B. Niethammer and J. Velasquez:
**Self-similar solutions for the LSW model with encounters,**

preprint (2008).

G. Menon, B. Niethammer and R.L. Pego:
**Dynamics and self-similarity in min-driven clustering,**

preprint (2008)