Directed polymer in random environnement: folding and location in dimension 1+1

I will present a one dimensional model for a polymer in a poor solvent: the (simple symmetric) random walk (RW) in dimension 1 penalized by its range in a random environement. We consider a random field $\omega$ consisting of iid variables $\omega_z$ and define the random Gibbs transformation of the simple symetric RW by a weight $\exp(-H_n)$, where $H_n$ is the hamiltonian and is the sum of $h - \beta \omega_z$ over all the visited sites $z$. The parameters $h, \beta$ are supposed to be nonnegative, meaning the polymer tends to fold itself on an optimal segment given by the random field.

In this talk, we will see how we can compute asymptotics for the partition function, defined as the renormalization quantity of the Gibbs measure, to deduce the polymer's typical behavior. We will do so in three disctinct settings:
- Homogeneous setting: suppose $\beta = 0$, then we can deduce scaling limits in distribution for the size and the location of the polymer.
- General setting: suppose $\beta > 0$, by studying the log-partition function we can deduce almost sure scaling limits "in probability" (under the Gibbs measure) for the size and the location of the polymer, as well as the fluctuations at a lower scale.
- Simplified model: we fix one of the edges of the polymer to simplify the calculations, then we can go further and possibly give the law of the fluctuations (still a work in progress)