Complex energy landscape with many metastable states is ubiquitous in many disordered
systems ranging from statistical physics to high dimensional statistics, machine learning, biology
In this talk we focus on a wide class of simple glassy models characterized by a class of random
Hamiltonians on the N1 dimensional sphere of radius square root of N. It is known recently
that these systems exhibit phase transitions in terms of the topology of energy landscape.
In particular, when the sigmaltonoise ratio is beyond some threshold, their ground states are
conjectured to be replicasymmetric which is instrumental in characterizing dynamical behavior.
We recently derive an explicit formula for their limiting ground states by combining techniques in spin glass theory and random matrices. Moreover we can characterize the mean number
of deep minima near the bottom of the landscape which shows some interesting phenomenon
beyond the physics picture onstage.
