We develop a theory for the market impact of large trading orders, which we call metaorders because they are typically split into small pieces and executed incrementally. Market impact is empirically observed to be a concave function of metaorder size, i.e. the impact per share of large metaorders is smaller than that of small metaorders. We formulate a stylized model of an algorithmic execution service and derive a fair pricing condition, which says that the average transaction price of the metaorder is equal to the price after trading is completed. We show that at equilibrium the distribution of trading volume adjusts to reflect information, and dictates the shape of the impact function. The resulting theory makes empirically testable predictions for the functional form of both the temporary and permanent components of market impact. Based on the commonly observed asymptotic distribution for the volume of large trades, it says that market impact should increase asymptotically roughly as the square root of metaorder size, with average permanent impact relaxing to about two thirds of peak impact.

We develop a theory for the market impact of large trading orders, which we call metaorders because they are typically split into small pieces and executed incrementally. Market impact is empirically observed to be a concave function of metaorder size, i.e. the impact per share of large metaorders is smaller than that of small metaorders. We formulate a stylized model of an algorithmic execution service and derive a fair pricing condition, which says that the average transaction price of the metaorder is equal to the price after trading is completed. We show that at equilibrium the distribution of trading volume adjusts to reflect information, and dictates the shape of the impact function. The resulting theory makes empirically testable predictions for the functional form of both the temporary and permanent components of market impact. Based on the commonly observed asymptotic distribution for the volume of large trades, it says that market impact should increase asymptotically roughly as the square root of metaorder size, with average permanent impact relaxing to about two thirds of peak impact. References and Notes 23A. Proofs of the propositions 27 B. Market impact for other metaorder size distributions: stretched exponential and lognormal 32 C. Effect of finite M on impact find concave temporary impacts roughly consistent with a square root functional form. The functional form of permanent impact is harder to measure and more controversial. These studies should be distinguished from the large number of studies of the market impact of individual trades or the sum of trades in a given period of time, that do not attempt to link together the individual trades coming from a given client

In the light of a recently derived evolution equation for genetic algorithms we consider the schema theorem and the building block hypothesis. We derive a schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The equation makes manifest the content of the building block hypothesis showing how fit schemata are constructed from fit sub-schemata. However, we show that, generically, there is no preference for short, low-order schemata. In the case where schema reconstruction is favored over schema destruction, large schemata tend to be favored. As a corollary of the evolution equation we prove Geiringer's theorem.

Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature, but with global degrees of freedom related to the topology of the universe. We present an exact Hamiltonian lattice theory for topological gravity, which admits translations of the lattice sites as a gauge symmetry. There are additional symmetries, not present in Einstein's theory, which kill the local degrees of freedom. We show that these symmetries can be fixed by choosing a gauge where the torsion is equal to zero. In this gauge, the theory describes flat space-times. We propose two methods to advance towards the holy grail of lattice gravity: A Hamiltonian lattice theory for curved space-times, with first-class translation constraints.

This is a theory of 2+1 gravity with a spacelike lattice and continuous time. It is derived from the Chern-Simons formulation of 2+1 gravity, in which the frame vectors are canonically conjugate to the SO(2, 1) connection. Dirac brackets are computed for the corresponding lattice variables: the link 3-vectors and the SO(2, 1) matrices which define parallel transport across the boundary between neighbouring faces. There is a set of first-class constraints, which state that the lattice analogues of torsion and curvature vanish locally. These generate local Lorentz rotations and independent translations of each lattice site. One finds the same complete set of observables as in the Chern-Simons formulation.