**Authors**
: Alessandro Giuliani, Joel L. Lebowitz, Elliott H. Lieb

**Title:
Pattern formation in systems with competing interactions
**

**Abstract**:
There is a growing interest, inspired by advances in technology, in the low
temperature physics of thin films. These quasi-2D systems show a
wide range of ordering effects including formation of striped states,
reorientation transitions, bubble formation in strong magnetic fields,
etc. The origins of these phenomena are, in
many cases, traced to competition between short ranged exchange
ferromagnetic interactions, favoring a homogeneous ordered state,
and the long ranged dipole-dipole interaction, which opposes such
ordering on the scale of the whole sample. The present theoretical
understanding of these phenomena is based on a combination of
variational methods and a variety of approximations, e.g., mean-field
and spin-wave theory. The
comparison between the predictions of these approximate methods and
the results of MonteCarlo simulations are often difficult because of
the slow relaxation dynamics associated with the long-range nature of
the dipole-dipole interactions. In this note we will review recent work
where we prove existence of periodic structures in some lattice and
continuum model systems with competing interactions. The continuum
models have also been used to describe micromagnets, diblock polymers, etc.

**Keywords**:
Striped order, periodic ground state,
Ising model, reflection positivity.

Alessandro Giuliani,

Dipartimento di Matematica,

Universita' di Roma Tre,

L.go S. Leonardo Murialdo 1, 00146 Roma, Italy

giuliani@mat.uniroma3.it

Joel L. Lebowitz,

Department of Mathematics and Physics,

Rutgers University, Rutgers University, Piscataway, NJ 08854 USA

lebowitz@math.rutgers.edu

Elliott H. Lieb,

Department of Mathematics and Physics,

Princeton University, Princeton 08544 NJ, USA

lieb@math.princeton.edu