Random Manhattan Research Articles

Tracer diffusion on a crowded random Manhattan lattice

We study, by extensive numerical simulations, the dynamics of a hard-core tracer particle (TP) in presence of two competing types of disorder—frozen convection flows on a square random Manhattan lattice and a crowded dynamical environment formed by a lattice gas of mobile hard-core particles. The latter perform lattice random walks, constrained by a single-occupancy condition of each lattice site, and are either insensitive to random flows (model A) or choose the jump directions as dictated by the local directionality of bonds of the random Manhattan lattice (model B). We focus on the TP disorder-averaged mean-squared displacement, (which shows a super-diffusive behaviour ∼t4/3, t being time, in all the cases studied here), on higher moments of the TP displacement, and on the probability distribution of the TP position X along the x-axis, for which we unveil a previously unknown behaviour. Indeed, our analysis evidences that in absence of the lattice gas particles the latter probability distribution has a Gaussian central part , where u = X/t2/3, and exhibits slower-than-Gaussian tails for sufficiently large t and u. Numerical data convincingly demonstrate that in presence of a crowded environment the central Gaussian part and non-Gaussian tails of the distribution persist for both models.

Matrix model for strings beyond thec=1barrier: The spin-sHeisenberg model on random surfaces

We consider a spin-s Heisenberg model coupled to two-dimensional quantum gravity. We quantize the model using the Feynman path integral, summing over all possible two-dimensional geometries and spin configurations. We regularize this path integral by starting with the R-matrices defining the spin-s Heisenberg model on a regular 2d Manhattan lattice. 2d quantum gravity is included by defining the R-matrices on random Manhattan lattices and summing over these, in the same way as one sums over 2d geometries using random triangulations in non-critical string theory. We formulate a random matrix model where the partition function reproduces the annealed average of the spin-s Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.

The XXZ Heisenberg model on random surfaces

We consider integrable models, or in general any model defined by an R-matrix, on random surfaces, which are discretized using random Manhattan lattices. The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. They can also be associated with the random graphs of multiparticle scattering nodes. As an example we formulate a random matrix model where the partition function reproduces the annealed average of the XXZ Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.

Exact exponent relations for random resistor-diode networks

Investigates the percolation properties of a random network of non-directed bonds (resistors) and arbitrarily oriented directed bonds (diodes). For the square lattice, there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes ('random Manhattan'). Along this line, symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant. Furthermore, a lattice independent relation is proved which shows that at the isotropic percolation threshold, the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes.