Sutapa Mukherji — Page 3

arXiv1993

Directed Polymers with Random Interaction : An Exactly Solvable Case -

We propose a model for two $(d+1)$-dimensional directed polymers subjected to a mutual $δ$-function interaction with a random coupling constant, and present an exact renormalization group study for this system. The exact $β$-function, evaluated through an $ε(=1-d)$ expansion for second and third moments of the partition function, exhibits the marginal relevance of the disorder at $d=1$, and the presence of a phase transition from a weak to strong disorder regime for $d>1$. The lengthscale exponent for the critical point is $ν=1/2\midε\mid$. We give details of the renormalization. We show that higher moments do not require any new interaction, and hence the $β$ function remains the same for all moments. The method is extended to multicritical systems involving an $m$ chain interaction. The corresponding disorder induced phase transition for $d>d_m=1/(m-1)$ has the critical exponent $ν_m=[2d(m-1)-2]^{-1}$. For both the cases, an essential singularity appears for the lengthscale right at the upper critical dimension $d_m$. We also discuss the strange behavior of an annealed system with more than two chains with pairwise random interactions among each other.

cond-mat↗

arXiv1993

Reunion of Vicious Walkers: Results from $ε$-Expansion -

The anomalous exponent, $η_{p}$, for the decay of the reunion probability of $p$ vicious walkers, each of length $N$, in $d$ $(=2-ε)$ dimensions, is shown to come from the multiplicative renormalization constant of a $p$ directed polymer partition function. Using renormalization group(RG) we evaluate $η_{p}$ to $O(ε^2)$. The survival probability exponent is $η_{p}/2$. For $p=2$, our RG is exact and $η_p$ stops at $O(ε)$. For $d=2$, the log corrections are also determined. The number of walkers that are sure to reunite is 2 and has no $ε$ expansion.

cond-mat↗