Fabio Sartori - Università di Parma #
Random walks on combs: Covering and Hitting times #
In this work we consider a simple random walk embedded on a 
two-dimensional regular comb and we address two, intrinsically related 
problems, i.e. the set of hitting times \(\{ H_{ij} \}\) and the 
covering 
time \(\tau\). As for the former, by exploiting the resistance method, 
we 
get analytically the exact expression for the set of hitting times, 
whose mean directly gives the global mean first passage time on combs. 
We also notice that the mean time to first reach any end-node of a side 
chain, starting from the backbone, scales as \(\sim L^3\). This turns 
out 
to be the leading term for the covering time, as shown via numerical 
simulations. Finally, we investigate the problem of "imperfect 
covering", where we look for the mean time \(\tau(x)\) such that a 
fraction \(x\) of the underlying structure has been covered. The growth 
of 
\(\tau(L,x) \approx (a x^{-2/3}-b)\log (L)\) suggests optimization 
strategies, as a confidence interval of \(1\%\) allows a drastic 
reduction 
of the covering time: \(\tau(2^{10},1) / \tau(2^{10},0.99) \approx 3\).