  We consider the model of the directed polymer in a random medium of dimension
1+3, and investigate its multifractal properties at the
localization/delocalization transition. In close analogy with models of the
quantum Anderson localization transition, where the multifractality of critical
wavefunctions is well established, we analyse the statistics of the position
weights $w_L(\vec r)$ of the end-point of the polymer of length $L$ via the
moments $Y_q(L) = \sum_{\vec r} [w_L(\vec r)]^q$. We measure the generalized
exponents $\tau(q)$ and $\tilde \tau(q)$ governing the decay of the typical
values $Y^{typ}_q(L) = e^{\bar{\ln Y_q(L)}} \sim L^{- \tau(q)} $ and
disorder-averaged values $\bar{Y_q(L)} \sim L^{- \tilde \tau(q)} $
respectively. To understand the difference between these exponents, $ \tau(q)
\neq \tilde \tau(q)$ above some threshold $q>q_c \sim 2$, we compute the
probability distributions of $y=Y_q(L)/Y^{typ}_q(L) $ over the samples : we
find that these distributions becomes scale invariant with a power-law tail
$1/y^{1+x_q}$. These results thus correspond to the Ever-Mirlin scenario [Phys.
Rev. Lett. 84, 3690 (2000)] for the statistics of Inverse Participation Ratios
at the Anderson localization transitions. Finally, the finite-size scaling
analysis in the critical region yields the correlation length exponent $\nu
\sim 2$.
