4,285 research outputs found
Pinning of polymers and interfaces by random potentials
We consider a polymer, with monomer locations modeled by the trajectory of a
Markov chain, in the presence of a potential that interacts with the polymer
when it visits a particular site 0. Disorder is introduced by, for example,
having the interaction vary from one monomer to another, as a constant plus
i.i.d. mean-0 randomness. There is a critical value of above which the
polymer is pinned, placing a positive fraction of its monomers at 0 with high
probability. This critical point may differ for the quenched, annealed and
deterministic cases. We show that self-averaging occurs, meaning that the
quenched free energy and critical point are nonrandom, off a null set. We
evaluate the critical point for a deterministic interaction ( without added
randomness) and establish our main result that the critical point in the
quenched case is strictly smaller. We show that, for every fixed
, pinning occurs at sufficiently low temperatures. If the
excursion length distribution has polynomial tails and the interaction does not
have a finite exponential moment, then pinning occurs for all
at arbitrary temperature. Our results apply to other mathematically similar
situations as well, such as a directed polymer that interacts with a random
potential located in a one-dimensional defect, or an interface in two
dimensions interacting with a random potential along a wall.Comment: Published at http://dx.doi.org/10.1214/105051606000000015 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Subgaussian concentration and rates of convergence in directed polymers
We consider directed random polymers in (d+1) dimensions with nearly gamma i.i.d. disorder. We study the partition function ZN,ω and establish exponential concentration of log ZN,ω about its mean on the subgaussian scale √N/log N . This is used to show that E[log ZN,ω] differs from N times the free energy by an amount which is also subgaussian (i.e. o(√N)), specifically O(√N/logN log log N)
Layering and wetting transitions for an SOS interface
We study the solid-on-solid interface model above a horizontal wall in three
dimensional space, with an attractive interaction when the interface is in
contact with the wall, at low temperatures. There is no bulk external field.
The system presents a sequence of layering transitions, whose levels increase
with the temperature, before reaching the wetting transition.Comment: 61 pages, 6 figures. Miscellaneous corrections and changes, primarily
in Section 4. Figure 5 added
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