Memcomputing is a novel computing paradigm that employs time non-locality
(memory) to solve combinatorial optimization problems. It can be realized in
practice by means of non-linear dynamical systems whose point attractors
represent the solutions of the original problem. It has been previously shown
that during the solution search digital memcomputing machines go through a
transient phase of avalanches (instantons) that promote dynamical long-range
order. By employing mean-field arguments we predict that the distribution of
the avalanche sizes follows a Borel distribution typical of critical branching
processes with exponent $\tau= 3/2$. We corroborate this analysis by solving
various random 3-SAT instances of the Boolean satisfiability problem. The
numerical results indicate a power-law distribution with exponent $\tau = 1.51
\pm 0.02$, in very good agreement with the mean-field analysis. This indicates
that memcomputing machines self-tune to a critical state in which avalanches
are characterized by a branching process, and that this state persists across
the majority of their evolution.

We carry out a comprehensive theoretical and experimental study of charge injection in poly(3-hexylthiophene) (P3HT) to determine the most likely scenario for metal-insulator transition in this system. Wecalculate the optical-absorption frequencies corresponding to a polaron and a bipolaron lattice in P3HT. We also analyze the electronic excitations for three possible scenarios under which a first- or a second-order metal-insulator transition can occur in doped P3HT. These theoretical scenarios are compared with data from infrared absorption spectroscopy on P3HT thin-film field-effect transistors (FETs). Our measurements and theoretical predictions suggest that charge-induced localized states in P3HT FETs are bipolarons and that the highest doping level achieved in our experiments approaches that required for a first-order metal-insulator transition.