DNA denaturation : a phase transition with disorder
Martin Retaux (LPS)

This thesis is a study of a DNA denaturation model, introduced by Poland
and Scheraga during the 1960s. The depinning models with random
environment, with which the similarity has been made, are also concerned.
If the interactions between the system and the environment are
homogeneous, the problem has been solved : depending on the value of a
geometrical parameter, a first or a second order phase transition happens.
On the other hand, when the interactions are random, we know neither the
critical point nor the phase transition order in the case of strong
disorder.
In order to simplify the problem, some authors have used a hierarchical
representation through which an exact renormalization can be written.
Despite this simplification, the critical point and the transition order
have not been found.
By changing the renormalization relation, we introduced a Toy-model which
is simpler than the hierarchical version. The new problem leaded us to a
family of distributions, which stay almost the same under renormalization,
and allow us to derive the Berezinskii-Kosterlitz-Thouless equations.
Also, with strong disorder, the phase transition does not have a critical
fixed point. These two elements, according to our numerical results,
predict that the order transition is infinite.
The second part of this thesis reports on a work about the simple
symmetric exclusion process, which is one of the simplest out of
equilibrium models for which a stationary state is known. The large
deviation function has been calculated in the past through microscopic and
macroscopic approaches. Here, we calculated the leading finite-size
correction. Then the result has been compared to similar corrections for
equilibrium systems.