Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices

Details Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices

2007-10-03

arxiv.org/abs/0710.0735v1

Phys. Rev. E 77, 021132 (2008)

Physics

Condensed Matter

Disordered Systems and Neural Networks

23 pages, 18 figures

Scientific paper

10.1103/PhysRevE.77.021132

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 $z=Z/Z_{typ}$,we find that the critical point corresponds to a fixed point distribution with a power-law tail $P_c(z) \sim \Phi(\ln z)/z^{1+\mu}$ as $z \to +\infty$ (up to some sub-leading logarithmic correction $\Phi(\ln z)$), so that all moments $z^{n}$ with $n>\mu$ diverge. For the wetting transition, the first moment diverges $\bar{z}=+\infty$ (case $0<\mu<1$), and the critical temperature is strictly below the annealed temperature $T_c

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