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__Preprint:__

Charged polymers in the attractive regime: a first order transition
from Brownian scaling to four points localization

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Yueyun Hu, Davar Khoshnevisan, and Marc Wouts

**Abstract.**
We study a quenched charged-polymer model, introduced by Garel and Orland
in 1988, that reproduces the folding/unfolding transition of biopolymers.
We prove that, below the critical inverse temperature,
the polymer is delocalized in the sense
that: (1) The rescaled trajectory of the polymer converges to the
Brownian path; and (2) The partition function remains bounded.
At the critical inverse temperature, we show that the maximum
time spent at points jumps discontinuously from 0 to a
positive fraction of the number of monomers, in the limit as the number of monomers
tends to infinity.
Finally, above the critical inverse temperature,
we prove that the polymer collapses in the sense
that a large fraction of its monomers live on four adjacent
positions, and its diameter grows only logarithmically
with the number of the monomers.
Our methods also provide some insight into the annealed phase
transition and at the transition due to a pulling force; both phase
transitions are shown to be discontinuous.
**Keywords.**
Charged polymers, quenched measure, annealed
measure, localization-delocalization transition, first-order phase
transition.

**
AMS Classification (2000)** Primary: 60K35; Secondary: 60K37.

**Support**. Research of D.K. was supported in part by the
National Science Foundation grants DMS-0706728 and DMS-1006903.

**Pre/E-Prints.** This paper (50 pages) is available in

Yueyun Hu
Départment de Mathématiques
Université Paris 13
99 avenue J-B Clément, F-93430,
Villetaneuse, France
yueyun@math.univ-paris13.fr |
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
davar@math.utah.edu |
Marc Wouts
Départment de Mathématiques
Université Paris 13
99 avenue J-B Clément, F-93430,
Villetaneuse, France
marc.wouts@math.univ-paris13.fr |

*Last Update: November 4, 2010*

© 2010 - Yueyun Hu, Davar Khoshnevisan, and Marc Wouts