Nested Critical Points for a Directed Polymer on a Disordered Diamond Lattice

We consider a model for a directed polymer in a random environment defined on a hierarchical diamond lattice in which i.i.d. random variables are attached to the lattice bonds. Our focus is on scaling schemes in which a size parameter n, counting the number of hierarchical layers of the system, becomes large as the inverse temperature $$\beta $$ β vanishes. When $$\beta $$ β has the form $${\widehat{\beta }}/\sqrt{n}$$ β ^ / n for a parameter $${\widehat{\beta }}>0$$ β ^ > 0 , we show that there is a cutoff value $$0 \kappa $$ β ^ > κ . We obtain a more refined description of the border between these two regimes by setting the inverse temperature to $$\kappa /\sqrt{n} + \alpha _n$$ κ / n + α n where $$0 \eta $$ α > η . Extending the analysis yet again by probing around the inverse temperature $$(\kappa / \sqrt{n}) + \eta (\log n-\log \log n)/n^{3/2}$$ ( κ / n ) + η ( log n - log log n ) / n 3 / 2 , we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases $${\widehat{\beta }}\le \kappa $$ β ^ ≤ κ and $$\alpha \le \eta $$ α ≤ η , this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.