The phase diagram of a directed polymer in random media with $p$-spin ferromagnetic interactions

We consider a directed polymer model with an additive $p$-spin ($p>2$) ferromagnetic term in the Hamiltonian. We give a rigorous proof for the specific free energy and derive the phase diagram. This model was proposed previously, and a detailed proof was given in the case $p=2$, while the main result was only stated for $p>2$. We give a detailed proof of the main result and show the behavior of the model as $p\to\infty$ by constructing the phase diagram also in this case. These results are important in many applications, for instance, in telecommunication and immunology. Our major finding is that in the phase diagram for $p>2$, a new transition curve (absent for $p=2$) emerges between the paramagnetic region and the so-called mixed region and that the ferromagnetic region diminishes as $p\to\infty$.