abstract:
For n a natural
number and r a positive number, the random geometric graph G(n,r) is
constructed by picking n points at random from d-dimensional space (iid
according to some probability distribution) and joining two points by an
edge if their distance is less than r. Typically one studies the behaviour
of G(n,r) as n grows, where r=r(n) is allowed to vary with n.
In this talk I will survey some of the most important results on this graph
model, including those on connectedness, the emergence of a "giant"
component, monotone properties and colouring.
abstract:
Transport of molecules
across cell membranes is an essential mechanism for life processes. These
molecules are often long and flexible, and the pores in the membranes are
too narrow to allow them to pass through as a single unit. In such
circumstances, the passage of a molecule through the pore --- translocation
--- proceeds through a random process in which polymer segments sequentially
move through the pore.
For statistical physicists, interesting quantities of translocation are
mainly scaling properties, for instance: how does the time it takes for a
polymer to translocate scale with its length? How long does a translocating
polymer block the pore during translocation, as a function of its length?
We will show that the translocation process can be viewed as a random-walk
process with anomalous diffusion, characterized by a squared displacement
scaling with time as $<r^2>
~ t^{(1+\nu)/(1+2\nu)}$, and that
the dwell time $\tau_d$
scales with polymer length as
$\tau_d ~ N^{2+\nu}$.
Here, $\nu \approx 0.588$
is the Flory exponent of self-avoiding walks. We will also discuss further
applications of the theoretical framework that yielded these results.