Prof. Buks van Rensburg

Professor E.J. Janse van Rensburg

Applied Mathematics Section
Faculty of Science and Engineering

OFFICIAL WEB SITE


Contact: Prof. E.J. Janse van Rensburg Department of Mathematics and Statistics York University Toronto, Ontario M3J 1P3 Canada

Phone:(416)-736-2100 X33837

Email:rensburg@yorku.ca

Research

· Statistical Mechanics

· Combinatorics

· Monte Carlo Simulations

· BioInformatics

· Statistical Knot Theory

COURSE WEBPAGES:

Math2041
Math6280

Selected Papers (Here are links to journals or pdf-files).

Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption. A. Rechnitzer and E.J. Janse van Rensburg. The Electronic Journal of Combinatorics. 9 (2002) R20.
Canonical Monte Carlo Determination of the Connective Constant of Self Avoiding Walks. A. Rechnitzer and E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. 35 (2002) L605-L612.
High Precision Canonical Monte Carlo Determination of the Growth Constant of Square Lattice Trees. E.J. Janse van Rensburg and A. Rechnitzer. Physical Review E. 67 (2003) 036116-1-9.
The Probability of Knotting in Lattice Polygons. E.J. Janse van Rensburg. Contemporary Mathematics.304 (2002) 125-135..
Knotting in Adsorbing Lattice Polygons. E. J. Janse van Rensburg. Contemporary Mathematics. 304 (2002) 137-151.
Statistical Mechanics of Directed Models of Polymers in the Square Lattice. E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. 36 (2003) R11-R61.
Exchange Relations, Dyck Paths and Copolymer Adsorption. A. Rechnitzer and E.J. Janse van Rensburg. Discrete Applied Mathematics. 140 (2004) 49-71.
Multiple Markov Chain Monte Carlo Study of Adsorbing Self Avoiding Walks in Two and Three Dimensions. E.J. Janse van Rensburg and A. Rechnitzer. Journal of Physics A: Math. Gen. 37 (2004) 6875-6898.
Inflating Square and Rectangular Lattice Vesicles. E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. 37 (2004) 3903-3932.
A Tutorial on Knot Energies. E.J. Janse van Rensburg. In Physical and Numerical Models in Knot Theory. Editors Calvo, Millett, Rawdon and Stasiak (2004) 19-44.
Rectangular Vesicles in Three Dimensions. J. Ma en E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. 38 (2005) 4115-4147.
Square Lattice Directed Paths Adsorbing on the Line Y=qX. E.J. Janse van Rensburg. Journal of Statistical Mechanics: Theory and Experiment. (September 2005) P09010
Forces in Square Lattice Directed Paths in a Wedge. E. J. Janse van Rensburg and Y. Le. Journal of Physics A: Math. Gen. 38 (2005) 8493-8503.
Adsorbing Bargraph Paths in a q-Wedge. E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. 38 (2005) 8505-8525. Corrigendum:Journal of Physics A: Math. Gen. (2006) 39 3847.
Self-avoiding Walks in a Slab with Attractive Walls. E.J. Janse van Rensburg, E. Orlandini, A.L. Owczarek, A. Rechnitzer, and S.G. Whittington. Journal of Physics A: Math. Gen. 38 (2005) L823-828.
Forces in Motzkin Paths in a Wedge. E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. 39 (2006) 1581-1608.
Moments of Directed Paths in a Wedge. E.J. Janse van Rensburg. Journal of Physics A: Math. Gen. Conf. Ser. 42 (2006) 147-162.
Plane Partition Vesicles. E.J. Janse van Rensburg and J. Ma. Journal of Physica A: Math. Gen. 39 (2006) 11171-11192.
Self-Avoiding Walks in a Slab: Rigorous Results. E.J. Janse van Rensburg, E. Orlandini and S.G. Whittington. Journal of Physics A: Math. Gen. 39 (2006) 13869-13902.
Partially Directed Paths in a Wedge. E.J. Janse van Rensburg, T. Prellberg and A.R. Rechnitzer. In Print. Journal of Combinatorial Theory, Serial A.
Squeezing Knots. E.J. Janse van Rensburg. Journal of Statistical Mechanics: Theory and Experiment. (March 2007) P03001.
Thoughts on Lattice Knot Statistics. E.J. Janse van Rensburg. Journal of Mathematical Chemistry 45(1) 7-38. (Commemorative issue in honour of Stu Whittington and Ray Kapral). *
Monte Carlo Methods for Lattice Polygons. E.J. Janse van Rensburg. In Polygons, Polyominoes and Polyhedra Ed. A.J. Guttmann. Canopus Publishing Ltd..
Directed Paths in a Wedge. E.J. Janse van Rensburg, T. Prellberg and A.R. Rechnitzer. Journal of Physics A: Math. Theor. 40 (2007) 14069 (16pp) .
Knot Probability of Polygons Subjected to a Force: A Monte Carlo Study. E.J. Janse van Rensburg, E. Orlandini, M.C. Tesi and S.G. Whittington. Journal of Physics A: Mathematical and Theoretical 41 (2008) 025003 (14pp).
Knotting in Stretched Polygons. E.J. Janse van Rensburg, E. Orlandini, M.C. Tesi and S.G. Whittington. Journal of Physics A: Mathematical and Theoretical 41 (2008) 015003 (25pp).
Atmospheres of Polygons and Knotted Polygons. E.J. Janse van Rensburg and A. Rechnitzer. Journal of Physics A: Mathematical and Theoretical 41 (2008) 105002 (23pp).
Self-avoiding Walks and Polygons in Slits. J. Alvarez, E.J. Janse van Rensburg, C.E. Soteros and S.G. Whittington. Journal of Physics. A: Mathematical and Theoretical 41 (2008) 185004 (22pp).
Directed Paths in a Layered Environment. J. Alvarez and E.J. Janse van Rensburg. Journal of Physics. A: Mathematical and Theoretical 41 (2008) 465003 (41pp)
Generalised Atmospheric Rosenbluth Methods (GARM). A. Rechnitzer and E.J. Janse van Rensbrug. Journal of Physics. A: Mathematical and Theoretical 41 (2008) 442002 (9pp).
Generalised Atmospheric Sampling of Self-Avoiding Walks. E.J. Janse van Rensburg and A. Rechnitzer. Journal of Physics. A: Mathematical and Theoretical 42 (2009) 335001 (29pp)
Approximate Enumeration of Self-Avoiding Walks. E.J. Janse van Rensburg. Preprint.
Monte Carlo Methods for the Self-Avoiding Walk. E.J. Janse van Rensburg. Journal of Physics. A: Mathematical and Theoretical 42 (2009) 323001 (97pp).
Thermodynamics and Entanglements of Walks under Stress. E.J. Janse van Rensburg, E. Orlandini, M.C. Tesi and S.G. Whittington. Journal of Statistical Mechanics: Theory and Experiment (July 2009) P07014 (35pp)
Atmospheric Collapse in Self-Avoiding Walks: A Numerical Study using GARM. J. Alvarez, M. Gara, E.J. Janse van Rensburg and A. Rechnitzer. Submitted to Journal of Statistical Mechanics:
Theory and Experiment

Random Maze

Here is a random maze I have generated by Monte Carlo. Can you find the way out?

In reality, a random maze would be much more convoluted than the above example. Such mazes are less pleasing to the eye. In the case above a maze consisting of walls that spiral from the origin was generated, and then subjected to a Monte Carlo simulation for a short time (before the maze would randomize completely).

Random Tree:

A maze is a special kind of random tree: in particular, it is a spanning tree of a square in the square lattice (such a maze would have only one way out from the center). Below is a random tree in the square lattice.

Towards the limiting random lattice tree

If the number of edges in the tree above should be multiplied, while the length scale is shrunk appropriately, a limiting random tree will be seen. In the picture below there are 10000 edges, each too small to be seen here, in a lattice tree generated by a Monte Carlo program. The tree begins to appear like a fractal object. The existence of a scaling limit is known in high dimensions (above 8). There is general consensus that it also exists in dimensions below 9.

Random Disks

The interior of a closed loop in the square lattice (or a polygon) is a Disk. In this example, a square lattice polygon was randomized by subjecting it to a Metropolis Monte Carlo algorithm. The interior of the polygon is a disk.

Die Oranje Vrystaat.

Die foto is in die Noord Vrystaat geneem.