On a Nielsen-Thurston classification theory for cluster modular groups

We classify elements of a cluster modular group into three types. We characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble. The characterization gives an analogue of the Nielsen-Thurston classification theory on the mapping class group of a surface.Comment: 33 pages, 9 figures. To appear in Annales de l'Institut Fourie

Matrix model criticality and resonant tunneling

We suggest that the Hermitian matrix models with resonant tunneling may exhibit novel criticality. Some features of the proposed criticality are explored. In particular, we argue that the new critical point is connected with the first-order transition.Comment: 11 pages, 6 figures. v2: References adde