KPZ modes in $d$-dimensional directed polymers

We define a stochastic lattice model for a fluctuating directed polymer in $d\geq 2$ dimensions. This model can be alternatively interpreted as a fluctuating random path in 2 dimensions, or a one-dimensional asymmetric simple exclusion process with $d-1$ conserved species of particles. The deterministic large dynamics of the directed polymer are shown to be given by a system of coupled Kardar-Parisi-Zhang (KPZ) equations and diffusion equations. Using non-linear fluctuating hydrodynamics and mode coupling theory we argue that stationary fluctuations in any dimension $d$ can only be of KPZ type or diffusive. The modes are pure in the sense that there are only subleading couplings to other modes, thus excluding the occurrence of modified KPZ-fluctuations or L\'evy-type fluctuations which are common for more than one conservation law. The mode-coupling matrices are shown to satisfy the so-called trilinear condition.