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The center-bound excitonic diffusion on dendrimers subjected to several typesof non-homogeneous funneling potentials, is considered. We first study themean-first passage time (MFPT) for diffusion in a linear potential withdifferent types of correlated and uncorrelated random perturbations. Increasingthe funneling force, there is a transition from a phase in which the MFPT growsexponentially with the number of generations $g$, to one in which it does solinearly. Overall the disorder slows down the diffusion, but the effect is muchmore pronounced in the exponential compared to the linear phase. When thedisorder gives rise to uncorrelated random forces there is, in addition, atransition as the temperature $T$ is lowered. This is a transition from ahigh-$T$ regime in which all paths contribute to the MFPT to a low-$T$ regimein which only a few of them do. We further explore the funneling within arealistic non-linear potential for extended dendrimers in which the dependenceof the lowest excitonic energy level on the segment length was derived usingthe Time-Dependent Hatree-Fock approximation. Under this potential the MFPTgrows initially linearly with $g$ but crosses-over, beyond a molecular-specificand $T$-dependent optimal size, to an exponential increase. Finally we considergeometrical disorder in the form of a small concentration of long connectionsas in the {\it small world} model. Beyond a critical concentration ofconnections the MFPT decreases significantly and it changes to a power-law orto a logarithmic scaling with $g$, depending on the strength of the funnelingforce.Comment: 13 pages, 9 figure

Disorder and funneling effects on exciton migration in treelike dendrimers