Speaker : Professor Yu Zhang (University of Colorado) Title : 1.Large deviations in the reinforced random walk model on trees 2.A singularity at the criticality for the free energy in percolation. Time : 2018-12-17 (Mon) 14:30 - 16:40 Place : Seminar Room 722, Institute of Mathematics (NTU Campus) Abstract: 1.Consider the linearly reinforced and the once-reinforced random walk models in the transient phase on trees. We show the large deviations for the upper tails for both models. We also show the exponential decay for the lower tail in the once-reinforced random walk model. However, the lower tail is in polynomial decay for the linearly reinforced random walk model. 2.Consider percolation on the triangular lattice. Let $\kappa(p)$ be the free energy at the zero field. We show that $\kappa(p)$ is not third differentiable. This answers affirmatively a conjecture, asked by Sykes and Essam (1964) that $\kappa(p)$ has a singularity at the critical $p_c$.