Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this paper we develop a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow, and analyze it using simulation, dimensional analysis, and theoretical tools based on mean field approximations. The model makes testable predictions for basic properties of markets, such as price volatility, the depth of stored supply and demand vs. price, the bid-ask spread, the price impact function, and the time and probability of filling orders. These predictions are based on properties of order flow and the limit order book, such as share volume of market and limit orders, cancellations, typical order size, and tick size. Because these quantities can all be measured directly there are no free parameters. We show that the order size, which can be cast as a nondimensional granularity parameter, is in most cases a more significant determinant of market behavior than tick size. We also provide an explanation for the observed highly concave nature of the price impact function. On a broader level, this work suggests how stochastic models based on zero-intelligence agents may be useful to probe the structure of market institutions. Like the model of perfect rationality, a stochastic-zero intelligence model can be used to make strong predictions based on a compact set of assumptions, even if these assumptions are not fully believable. Contents

We study nonequilibrium phase transitions in a mass-aggregation model which allows for diffusion, aggregation on contact, dissociation, adsorption, and desorption of unit masses. We analyze two limits explicitly. In the first case mass is locally conserved, whereas in the second case local conservation is violated. In both cases the system undergoes a dynamical phase transition in all dimensions. In the first case, the steady state mass distribution decays exponentially for large mass in one phase, and develops an infinite aggregate in addition to a power-law mass decay in the other phase. In the second case, the transition is similar except that the infinite aggregate is missing. [S0031-9007(98)07462-6]

We propose a new model of self-organized criticality. A particle is dropped at random on a lattice and moves along directions specified by arrows at each site. As it moves, it changes the direction of the arrows according to fixed rules. On closed graphs these walks generate Euler circuits. On open graphs, the particle eventually leaves the system, and a new particle is then added. The operators corresponding to particle addition generate an abelian group, same as the group for the Abelian Sandpile model on the graph. We determine the critical steady state and some critical exponents exactly, using this equivalence.

We study the nonequilibrium phase transition in a model of aggregation of masses allowing for diffusion, aggregation on contact and fragmentation. The model undergoes a dynamical phase transition in all dimensions. The steady state mass distribution decays exponentially for large mass in one phase. On the contrary, in the other phase it has a power law tail and in addition an infinite aggregate. The model is solved exactly within a mean field approximation which keeps track of the distribution of masses. In one dimension, by mapping to an equivalent lattice gas model, exact steady states are obtained in two extreme limits of the parameter space. Critical exponents and the phase diagram are obtained numerically in one dimension. We also study the time dependent fluctuations in an equivalent interface model in (1+1) dimension and compute the roughness exponent χ and the dynamical exponent z analytically in some limits and numerically otherwise. Two new fixed points of interface fluctuations in (1 + 1) dimension are identified. We also generalize our model to include arbitrary fragmentation kernels and solve the steady states exactly for some special choices of these kernels via mappings to other solvable models of statistical mechanics.

We provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in non-equilibrium systems in the steady state, from the measurements of arbitrary current fluctuations. Our results are built upon the finite-time generalization of the thermodynamic uncertainty relation, and require only very short time series data from experiments. We illustrate our results with exact and numerical solutions for two colloidal heat engines. arXiv:1910.00476v2 [cond-mat.stat-mech]