Frank Redig

Research

The general themes of my research are mathematical statistical
physics and probability theory.
Generally speaking statistical physics explains macroscopic
phenomena such as phase transitions and transport phenomena from the microscopic
world of interacting atoms.
Below I describe some specific research themes.
Preprints of most of my papers can be found on the archive
http://xxx.lanl.gov in the section mathematical physics.
A (not yet updated) list of publications, can be found here

1. Self-organized criticality and abelian sandpiles

Since its introduction in 1987, the Bak-Tang-Wiesenfeld (BTW) sandpile-model of
self-organized criticality has attracted a lot of interest. In the combinatorics community it is known
under the name ``chip-firing".
The model is very easy to describe: consider a finite subset V of e.g. the two-dimensional
lattice Z^2. To each site x we associate a height h, an integer between 1 and 4 (number
of grains at site x).
We pick a random site and add one grain to that site. If h+1 > 4, then the site
``topples", giving one grain to each neighbor inside V, and loosing itself four grains.
A site on the boundary looses grains when it topples.
If we continue this process, then the system will end up in a stationary state, which is
the uniform measure on the unique recurrent class. Recurrent configurations can be caracterized
by the so-called ``burning algorithm" invented by Dhar. They are in one-to-one correspondence
with rooted spanning trees. Pointwise addition and relaxation of recurrent configurations
defines an abelian group, the so-called ``sandpile group of the graph" (in this case Z^2).
Numerical studies and exact
results indicate that in the stationary state of this model, the correlation between height variables
decays as a power, just as for an equilibrium system at the critical point.
Therefore BTW called this phenomenon ``self-organized criticality".
Our research on this subject tries to answer the following questions:
1) Do the stationary measures in finite volume converge (in the thermodynamic limit)
to a unique measure on
infinite volume height configurations ?
2) Can we define a Markov process on infinite volume height-configurations with this
measure as a unique stationary measure?
3) What are the ergodic properties of this Markov process?
4) Can we say something about finiteness and -more ambitiously- statistics of avalanches ?
5) What can we say about the abelian group structure in infinite volume ?

These questions are beyond the scope of standard constructions of infinite volume
limits of interacting Markov processes by means of the Hille-Yoshida theorem
as is done in classical interacting particle systems. Adding a grain at a particular site can possibly
influence sites very far away, and this makes questions 2-3 interesting and non-trivial. In particular,
the limiting process will in general not have the Feller property.

Some papers related to this subject:

1) ``On the thermodynamic limit for a one-dimensional sandpile process" (with
C. Maes, E. Saada and A. Van Moffaert)
Markov Proc. Rel. Fields 6, 1-21 (2000).
2) ``The abelian sandpile on an infinite tree" (with C. Maes and E. Saada)
Ann. Prob. 30, 2081-2107 (2002).
3) ``The abelian sandpile model, a mathematical introduction" (with R. Meester
and D. Znamenski), Markov Proc. and Rel. Fields, 7, 509-523 (2002).
4) ``The infinite volume of dissipative abelian sandpiles" (with C. Maes and E. Saada),
Comm math Phys 244 , 395--417 (2004).
5) ``Infinite volume limits of high dimensional sandpiles"
(with Antal Jarai, to appear in Prob. Th. and Rel. Fields)
6) Les Houches lecture notes 2005

A very nice computer experiment for the BTW model can be found on the
website:
http://www.cmth.bnl.gov/~maslov/Sandpile.htm

Here are some pictures and links to related problems
- Configuration obtained by repeatedly adding at the origin, starting from the empty configuration
- Goldbugs: http://www.math.wisc.edu/~propp/rotor-router-1.0/
-Nice paper on goldbugs: http://people.brandeis.edu/~kleber/Papers/rotor.pdf

2. Entropy production

In the study of systems driven away from equilibrium, a central notion is that of
entropy production (EP). Non-equilibrium steady states are typically current
carrying, implying that they are non-reversible. We study non-equilibrium steady
states in the context of interacting stochastic particle systems, such as the
asymmetric exclusion process. The EP is introduced as a function of the trajectory
of the system, indicating how far the system is driven away from equilibrium.
Its expected value (MEP) is the relative entropy (density) between the process
run forward and backwards in time. Positivity is an immediate consequence, and
for some spatially extended Markov processes we can show that MEP=0 is equivalent with
reversibility (``no symmetry breaking of time-reversal"). A symmetry in the large
deviations of EP similar to the Gallavotti-Cohen fluctuation theorem is obtained
naturally in this framework.

Below some papers related to the subject:

1. ``Positivity of entropy production" (with C. Maes)
J. Stat. Phys. 101, 3-15 (2000)
2.``Entropy production for interacting particle systems" (with C. Maes and M. Verschuere)
Markoc Proc. and Rel. Fields, 7, 119-134
3. ``On the definition of entropy production via examples" (with C. Maes and A. Van Moffaert)
J. Math Phys. 41, 1528-1554 (2000).
4. "From global to local fluctuation theorems" (with C. Maes and M. Verschuere)
Moscow Math. Journal, 1, 421-438 (2002).

3. Transformations of Gibbs measures and generalized Gibbs formalism

Since the discovery of the Griffiths-Pearce singularities, the rigorous
implementation of the renormalization group transformation (RGT) became
a non-trivial mathematical problem. If we apply a transformation T
(such as decimation, majority vote, block-spin average) to a Gibbs measure
then the transformed measure can be non-Gibbs (in the
sense of the classical formalism), i.e., it does not have a reasonable Hamiltonian.
In '95 Dobrushin proposed the idea of extending the classical
Gibbs formalism in order
to include such transformed measures. The basic idea is
to replace the uniformity of the convergence of the Hamiltonian by
almost sure absolute convergence (i.e., with probability one
with respect to the transformed measure).
Two questions then arise
1. Prove that the transformed measures of the form have an almost surely convergent
interaction.
2. Restore the classical thermodynamic formalism (variational principle)
in this context.

Applications of weakly Gibbsian measures arise in the context of
non-uniformly hyperbolic dynamical systems
(models of intermittency) such as
the Manneville-Pomeau map and in hidden Markov models. A new development where non-Gibbsian
measures show up is in the time-evolution of a Gibbs measure under a dynamics which has another
Gibbs measure as equilibrium. E.g. during the ``heating" of a low-temperature system (see reference 4).

Some papers dealing with this subject:

1. ``The restriction of the Ising model to a layer" (with C. Maes and A. Van Moffaert),
J. Stat. Phys. 94,893-912 (1999).
2. ``Percolation, path large deviations and weak Gibbsiannity" (with C. Maes, S. Shlosman
and A. Van Moffaert), Comm. Math. Phys. 209, 517-545, (2000).
3. ``Almost versus weakly Gibbsian measures" (with C. Maes and A. Van Moffaert),Stoch. Proc. Appl. 79, 1-15 (1999).
4.``Possible loss and recovery of Gibsianness during the stochastic evolution of Gibbs
measures" (with A.C.D. Van Enter, R. Fernandez and F. den Hollander),
Comm. Math. Phys. 226, 101-130 (2002).
5.``Intermittency and weak Gibbs states" (with C. Maes, A. Van Moffaert, F. Takens
and E. Verbitskiy), Nonlinearity 13, 1681-1698 (2000).

Other subjects of current research and interest:
- Exponential laws for hitting, waiting and return times : see e.g. exponential laws, appeared
in Comm Math Phys. 246, 269-294 (2004).
- Extreme values and distribution of maximal clusters: see e.g. this paper
- Large deviations questions for KMS-states of quantum spin systems: e.g. qrv
- Coupled map lattices
- Deviation and concentration inequalities for dependent random fields: see e.g. deviation