IOP PUBLISHING JOURNAL OF PHYSICS A:MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 46 (2013) 505002 (13pp) doi:10.1088/1751-8113/46/50/505002 Exactsolutionofananisotropic2Drandomwalk modelwithstrongmemorycorrelations JCCressoni 1, 2 ,GMViswanathan 3 andMAAdaSilva 1 1 Departamento de F´ ısica e Qu´ ımica, FCFRP, Universidade de S˜ ao Paulo, Ribeir˜ ao Preto, SP, 14040-903, Brazil 2 Instituto de F´ ısica, Universidade Federal de Alagoas, Macei´ o, AL, 57072-970, Brazil 3 Departamento de F´ ısica Te´ orica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-900, Brazil E-mail: maasilva@fcfrp.usp.br Received 27 August 2013 Published 26 November 2013 Online at stacks.iop.org/JPhysA/46/505002 Abstract Over the last decade, there has been progress in understanding one-dimensional non-Markovian processes via analytic, sometimes exact, solutions. The extension of these ideas and methods to two and higher dimensions is challenging. We report the first exactly solvable two-dimensional (2D) non- Markovian random walk model belonging to the family of the elephant random walk model. In contrast to L´ evy walks or fractional Brownian motion, such modelsincorporatememoryeffectsbykeepinganexplicithistoryoftherandom walk trajectory. We study a memory driven 2D random walk with correlated memory and stops, i.e. pauses in motion. The model has an inherent anisotropy with consequences for its diffusive properties, thereby mixing the dominant regime along one dimension with a subdiffusive walk along a perpendicular dimension. The anomalous diffusion regimes are fully characterized by an exact determination of the Hurst exponent. We discuss the remarkably rich phase diagram, as well as several possible combinations of the independent walks in both directions. The relationship between the exponents of the first and second moments is also unveiled. PACS numbers: 05.40.−a, 05.70.Ln, 02.50.Ey (Some figures may appear in colour only in the online journal) 1.Introduction Diffusionregimesaresaidtobeanomalouswhenthemean-square-displacement(MSD)grows asymptotically with time in a nonlinear fashion. In other words, if the MSD scales as t 2H , anomalous diffusion is defined by H �= 1/2, with superdiffusion associated with H > 1/2 and subdiffusion with H < 1/2. The ubiquitous normal diffusion is defined by H = 1/2 with a Gaussian random walk propagator. Systems with long-range memory correlations exhibiting 1751-8113/13/505002+13$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1