arXiv subjects

R. D. Morris

Publications and source records attributed to R. D. Morris.

3 recordsLinked to original sources

arXiv2004

Approximating satisfiability transition by suppressing fluctuations

Using methods and ideas from statistical mechanics, we propose a simple method for obtaining rigorous upper bounds for satisfiability transition in random boolean expressions composed of N variables and M clauses with K variables per clause. Determining the location of satisfiability threshold $α_c=M/N$ for a number of difficult combinatorial problems is a major open problem in the theory of random graphs. The method is based on identification of the core -- a subexpression (subgraph) that has the same satisfiability properties as the original expression. We formulate self-consistency equations that determine macroscopic parameters of the core and compute an improved annealing bound. We illustrate the method for three sample problems: K-XOR-SAT, K-SAT and positive 1-in-K-SAT.

cond-mat.dis-nn↗

arXiv2004

Quantum adiabatic optimization and combinatorial landscapes

In this paper we analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, $γ=M/N$. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (versus only energy) is used, and are able to show the existence of a dynamic threshold $γ=γ_d$ starting with some value of K -- the number of variables in each clause. Beyond dynamic threshold, the algorithm should take exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz. We have been able to map the ensemble of random graphs onto another ensemble with fluctuations significantly reduced. This enabled us to obtain tight upper bounds on satisfiability transition and to recompute the dynamical transition using the extended set of landscapes.

quant-ph↗

arXiv2002

Variable-Range Hopping of Spin Polarons: Magnetoresistance in a Modified Mott Regime

We analize electrical conductivity controlled by hopping of bound spin polarons in disordered solids with wide distributions of electron energies and polaron shifts (barriers). By means of percolation theory and Monte Carlo simulations we have shown that in such materials at low temperatures, when hopping occurs in the vicinity of the Fermi level, a hard polaron gap does not manifest itself in the transport properties. This happens because as temperature decreases the hopping polaron trades the decreasing electron and polaron barriers for increasing hopping distance. As a result, in the absence of the Coulomb correlation effects, in this variable-range variable-barrier hopping regime, the electrical resistivity as a function of temperature obeys a non-activation law, which differs from the standard Mott law.

cond-mat.mtrl-sci↗