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We provide a model of a decentralized, dynamic auction market platform (e.g., eBay) in which a large number of buyers and sellers participate in simultaneous, singleunit auctions each period. Our model accounts for the endogenous entry of agents and the impact of intertemporal optimization on bids. Solving our model with a finite number of bidders is computationally intractable due to the curse of dimensionality, so we prove that a continuum version of our model provides a good approximation of an equilibrium in the finite model. We use the approximation to estimate the structural primitives of our model using Kindle sales on eBay. We find that just over one third of Kindle auctions on eBay result in an inefficient allocation with deadweight loss amounting to 13.5% of total possible market surplus. We also find that partial centralization - for example, runnng half as many 2-unit, uniform price auctions each day - would eliminate a large fraction of the inefficiency, but yield lower seller revenues. Our results highlight the importance of understanding platform composition effects—selection of agents into the market—in assessing the implications of market design.

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We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

While two-dimensional symmetry-enriched topological phases (𝖲𝖤𝖳s) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by 𝖦𝖳) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (𝖲𝖤𝖦), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on 𝖲𝖤𝖦s with gauge group Gg=ℤN1×ℤN2×⋯ and on-site unitary symmetry group Gs=ℤK1×ℤK2×⋯ or Gs=U(1)×ℤK1×⋯. Each 𝖲𝖤𝖦(Gg,Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., 𝖲𝖤𝖳 orders) of 𝖲𝖤𝖦s in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the \emph{mixed} multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from 𝖲𝖤𝖦s to 𝖲𝖤𝖳s. By giving full dynamics to background gauge fields, 𝖲𝖤𝖦s may be eventually promoted to a set of new gauge theories (denoted by 𝖦𝖳∗). Based on their gauge groups, 𝖦𝖳∗s may be further regrouped into different classes each of which is labeled by a gauge group G∗g. Finally, a web of gauge theories involving 𝖦𝖳, 𝖲𝖤𝖦, 𝖲𝖤𝖳 and 𝖦𝖳∗ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.