Equality of critical points for polymer depinning transitions with loop exponent one. (English) Zbl 1187.82054

Alexander, Kenneth S.; Zygouras, Nikos

Ann. Appl. Probab. 20, No. 1, 356-366 (2010).

Summary: We consider a polymer with configuration modelled by the trajectory of a Markov chain, interacting with a potential of form \(u+V_n\) when it visits a particular state 0 at time \(n\), with \(\{V_n\}\) representing i.i.d. quenched disorder. There is a critical value of \(u\) above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length \(n\) takes the form \(\varphi (n)/n\) for some slowly varying \(\varphi \); this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of \(u\) in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ, at least at low temperatures.

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D60	Statistical mechanics of polymers
60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B27	Critical phenomena in equilibrium statistical mechanics

Keywords:

pinning; polymer; disorder; random potential; quenched critical point

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