2007/10/03 by Cécile Monthus, Cecile Monthus, Thomas Garel · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #Material Dynamics and Properties #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn
paper · pdf · doi:10.1103/physreve.77.021132
published as Phys. Rev. E 77, 021132 (2008) · 23 pages, 18 figures
arxiv created 2007/10/03 · openalex publication_date 2008/02/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the wetting transition and the directed polymer delocalization transition on diamond hierarchical lattices. These two phase transitions with frozen disorder correspond to the critical points of quadratic renormalizations of the partition function. (These exact renormalizations on diamond lattices can also be considered as approximate Migdal-Kadanoff renormalizations for hypercubic lattices.) In terms of the rescaled partition function z=Z/Ztyp, we find that the critical point corresponds to a fixed point distribution with a power-law tail Pc(z)\ensuremath∼\ensuremathΦ(ln z)/z^1+\ensuremathμ as z\ensuremath→+\ensuremath∞ [up to some subleading logarithmic correction \ensuremathΦ(ln z)], so that all moments zn with n>\ensuremathμ diverge. For the wetting transition, the first moment diverges z=+\ensuremath∞ (case 0<\ensuremathμ<1), and the critical temperature is strictly below the annealed temperature Tc<Tann. For the directed polymer case, the second moment diverges z2=+\ensuremath∞ (case 1<\ensuremathμ<2), and the critical temperature is strictly below the exactly known transition temperature T2 of the second moment. We then consider the correlation length exponent \ensuremathν: the linearized renormalization around the fixed point distribution coincides with the transfer matrix describing a directed polymer on the Cayley tree, but the random weights determined by the fixed point distribution Pc(z) are broadly distributed. This induces some changes in the traveling wave solutions with respect to the usual case of more narrow distributions.