Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices
C\ ' { e} cile Monthus and Thomas Garel

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 $n&gt\mu$ diverge. For the wetting transition, the first moment diverges $\overline{z}=+\infty$ (case $0&lt\mu&lt1$), and the critical temperature is strictly below the annealed temperature $T_c&ltT_{ann}$. For the directed polymer case, the second moment diverges $\overline{z^2}=+\infty$ (case $1&lt\mu&lt2$), and the critical temperature is strictly below the exactly known transition temperature $T_2$ of the second moment. We then consider the correlation length exponent $\nu$ : 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 $P_c(z)$ are broadly distributed. This induces some changes in the travelling wave solutions with respect to the usual case of more narrow distributions.

© 2008 The American Physical Society.