Abstract. Spectral properties of the Schrödinger operator on a finite compact metric graph with delta-type vertex conditions are discussed. Explicit estimates for the lowest eigenvalue (ground state) are obtained using two different methods: Eulerian cycle and symmetrization techniques. In the case of positive interactions even estimates for higher eigenvalues are derived.

It is common to model learning in games so that either a deterministic process or a finite state Markov chain describes the evolution of play. Such processes can however produce undesired outputs, where the players' behavior is heavily influenced by the modeling. In simulations we see how the assumptions in Young (1993), a well-studied model for stochastic stability, lead to unexpected behavior in games without strict equilibria, such as Matching Pennies. The behavior should be considered a modeling artifact. In this paper we propose a continuous-state space model for learning in games that can converge to mixed Nash equilibria, the Recency Weighted Sampler (RWS). The RWS is similar in spirit Young's model, but introduces a notion of best response where the players sample from a recency weighted history of interactions. We derive properties of the RWS which are known to hold for finite-state space models of adaptive play, such as the convergence to and existence of a unique invariant distribution of the process, and the concentration of that distribution on minimal CURB blocks. Then, we establish conditions under which the RWS process concentrates on mixed Nash equilibria inside minimal CURB blocks. While deriving the results, we develop a methodology that is relevant for a larger class of continuous state space learning models.