Distribution of Attraction Basins in a Family of Simple Glasses

We study the distribution of attraction basins as a function of energy in simple glasses. We find that it is always broad. Furthermore we identify two types of glasses, both with an exponentially large number of metastable states. In one type the largest attraction basin is exponentially small, whereas in the other it is polynomially small in the system size N. If there exists a tuning parameter that connects one regime with another, then these two phases are separated by a critical point. We discuss implications for optimization problems.