Statistics Seminar, Janos Englander (UC Boulder)
Tree builder random walks
We investigate a self-interacting random walk, in a dynamically evolving environment, which is a random tree built by the walker itself, as it walks around.
At time n=1,2,…, right before stepping, the walker adds a random number (possibly zero) Zn of leaves to its current position. We assume that the Zn's are independent but we do not assume that they are identically distributed.
We obtain non-trivial conditions on their distributions under which the random walk is recurrent.
This result is in contrast with previous work in which, under a sort of uniform ellipticity condition, namely, that infn P(Zn >=1)=kappa>0, the random walk was shown to be ballistic.
We also obtain results on the transience of the walk, and the possibility that it ``gets stuck.''
From the perspective of the environment, we provide structural information about the sequence of random trees generated by the model when Zn~Bernoulli(pn), with pn=Theta(n-ß ) where ß in (2/3,1].
Interestingly, the empirical degree distribution of this random tree sequence converges almost surely to a power-law distribution of exponent 3, thus revealing a connection to the well-known preferential attachment model of Barabasi-Albert.
This is joint work with R. Ribeiro (Boulder, USA) and G. Iacobelli (Rio de Janeiro, Brazil).
No particular background is needed.
Tid: 2022-06-17 13:15 till 14:00
s [dot] volkov [at] maths [dot] lth [dot] se