Firm-quasi-stability and re-equilibration in matching markets with contracts

We study firm-quasi-stability in the framework of many-to-many matching with contracts under substitutable preferences. We establish various links between firm-quasi-stability and stability and give new insights into the existence and lattice property of stable allocations. Moreover, we show that firm-quasi-stable allocations appears naturally when the stability of the market is disrupted by the entry of new firms or the retirement of some workers, and introduce a generalized deferred acceptance algorithm to show that the market can regain stability from firm-quasi-stable allocations by a decentralized process of offers and acceptances.Comment: 30 page

Holographic coherent states from random tensor networks

Random tensor networks provide useful models that incorporate various important features of holographic duality. A tensor network is usually defined for a fixed graph geometry specified by the connection of tensors. In this paper, we generalize the random tensor network approach to allow quantum superposition of different spatial geometries. We set up a framework in which all possible bulk spatial geometries, characterized by weighted adjacent matrices of all possible graphs, are mapped to the boundary Hilbert space and form an overcomplete basis of the boundary. We name such an overcomplete basis as holographic coherent states. A generic boundary state can be expanded on this basis, which describes the state as a superposition of different spatial geometries in the bulk. We discuss how to define distinct classical geometries and small fluctuations around them. We show that small fluctuations around classical geometries define "code subspaces" which are mapped to the boundary Hilbert space isometrically with quantum error correction properties. In addition, we also show that the overlap between different geometries is suppressed exponentially as a function of the geometrical difference between the two geometries. The geometrical difference is measured in an area law fashion, which is a manifestation of the holographic nature of the states considered.Comment: 33 pages, 8 figures. An error corrected on page 14. Reference update