Phase transition for random walks with attractive
interaction
Random walk is used by physicists to model polymer chains and interface of
two different substances. In these situations, the random walk
usually evolves
under influence of an exterior random object (e.g., a polymer chain in
a solvent which contains inpurity, or collection of polymer chains which
attract each other). Let us now suppose that the strength of the exterior
random object is measured by a parameter $\beta \ge 0$.
We are interested in the phase transition described as follows;
-
If $\beta$ is smaller than a certain critical
value $\beta_c >0$, then the presence
of the exterior random object does not change the large
time behavior of the random walk in a qualitative way (subcritical phase).
-
If $\beta$ is larger than the
critical value $\beta_c>0$, then the random walk
behaves quite differently from the usual one (supercritical phase).
We first try to find out exactly
when the difference occurs (the value of $\beta_c$)
by looking at the singularity
of the free energy. We then proceed to establish limit theorems
to describe the large time
behavior of the random walk in both subcritical and supercritical phase.
My coworkers in this line of research include Francis Comets,
Yasuki Isozaki, Tokuzo Shiga, and Hideki Tanemura.
References
-
(with F. Comets)
Some new results on Brownian directed polymers in random environment,
preprint (2004)
Download the ps.gz file/
Download the pdf file/
-
(with F. Comets and T. Shiga)
Probabilistic analysis of
directed polymers in random environment: a review,
to appear in
Advanced Studies in Pure Mathematics (2004)
Download the ps.gz file
-
(with F. Comets)
Brownian directed polymers in random environment,
preprint (2003)
Download the ps.gz file
-
(with F. Comets and T. Shiga)
Directed polymers in random environment: strong disorder case,
preprint (2002).
See the abstract. /
Download the dvi file
-
(with H. Tanemura) Localization transition of d-friendly walkers,
preprint (2001).
See the abstract. /
Download the dvi file
-
(with Y. Isozaki) Weakly pinned random walk on the wall: pathwise
descriptions of the phase transition,
Stoch. Proc. Appl. 96, 261 --284,(2001),
See the abstract. /
Download the dvi file