Guido Uguzzoni - Università degli Studi di Parma #
The true reinforced random walk with bias #
Stochastic processes with memory are a common way to model systems 
ranging from physics to ecology and biology. </br>
 In particular, self-attracting and self-avoiding random walks provide 
basic, yet non trivial, examples. </br>
Here, we first review the main definitions of  random walks with memory 
and then we focus on the case of (true) reinforced random walk where 
memory effects are implemented at each time step, differently from the 
static case, where memory effects are accounted for globally.</br>
We investigate the model in dimension $d=1$, also accounting for the 
presence of a field of strength $s$, which biases the walker toward a 
target site.
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We analyze in details the asymptotic long-time behavior of the walker 
through the main statistical quantities (e.g. distinct sites visited, 
end-to-end distance) and we discuss a possible mapping between such 
dynamic self-attracting model and the trapping problem for a simple random 
walk, in analogy with the static model. Moreover, we find that, for any 
$s>0$, the random walk behavior switches to ballistic and that field 
effects always prevail on memory effects without any singularity, already 
in $d=1$; this is in contrast with the behavior observed in the static 
model.