Condensed Matter

Tile Tool	A program tool I wrote to allow me to practice making some virtual tilings.
Jan 15, 2006	A progress report which overviews my work towards constructing an algorithm to count square-rhombus tilings on a torus.
Basic Path Counter	A dynamic-programming implementation of the basic path counter, described in the Jan 15 progress report.
Basic Path Counter Code	The code for the basic path counter program.
Advanced Path Counter	A dynamic-programming implementation of the advanced path counter, described in the Jan 15 progress report.
Advanced Path Counter Code	The code for the advanced path counter program.
Jan 24, 2006	Questions and information about the material I read and a simple proof.
Gessel-Viennot Algorithm	An implementation of the Gessel-Viennot Algorithm in Mathematica, which duplicates results from my Basic Path Counter.
Jan 31, 2006	The long-term behavior proof from last week applied to the Gessel-Viennot Algorithm's long-term results. Surprisingly, the paths seem to be almost oblivious to one another.
Gessel-Viennot Algorithm Long-Term	The companion notebook for January 31. A number of graphs illustrate how slight the sensitivity to other paths is.
Gessel-Viennot Algorithm Long-Term REVISION 2	The previous notebook had a few typos and was somewhat difficult to understand. This one clarifies a bit and collapses to 2D paths.
3 + sqrt(2) by 2 + sqrt(2) 3 + sqrt(2) by 2 + sqrt(2) #2 4 + 3sqrt(2) by 3 + 2sqrt(2) 4 + 4sqrt(2) by 3 + 2sqrt(2) 5 + sqrt(2) by 3 + 2sqrt(2) 5 + 2sqrt(2) by 3 + 2sqrt(2) 2 + sqrt(2) by 2	Sample tiled torii
Gessel-Viennot Algorithm Long-Term #3	Examining the long-term behavior of the entropy per unit as the length of the tail grows.
Feb 7, 2006	Explanation of the Gessel-Viennot Algorithm, and GVLongterm3 and squaregas.
Square Gas Experiments	An experiment involving repulsion from hard walls and an implementation of Destainville's algorithm (paper references inside)
Feb 14, 2006	Proof of the Gessel-Viennot algorithm, also pointing out the cases where it fails.
Gessel-Viennot Algorithm Fail Example	A Mathematica example showing a case where the algorithm fails.
Feb 16, 2006	Numerical analysis of the rate of convergence of entropies to the non-interacting limit.
Gessel-Viennot Algorithm Long-Term #4	Compainion notebook for Feb. 16
Gessel-Viennot Algorithm Long-Term #5	Again allowing path length and separation to vary, I graph entropy versus the parameter given slopes and path count.
Feb 21, 2006	A derivation looking for a deviation from the independent long-term Gessel-Viennot results for multiple paths.
Gessel-Viennot Algorithm Long-Term #6	Companion notebook for Feb. 21
Feb 28, 2006	The beginnings of a simple proof to Gessel-Viennot, and an attempt to extend last week's approximation to a periodic case. It looks so beautiful in its generating-function form, but it appears that the propagation of the edge effects extends irrevocably aclong the entire array of paths.
Gessel-Viennot Algorithm Long-Term #7	Companion notebook for Feb. 28
Mar 3, 2006	I simplified further the result from Feb 28, also extending it to arbitary displacement D. An approximation for paths>>length is developed, and an approximation for paths=length is suggested for a special case, and seems to work (shown in companion notebook).
Gessel-Viennot Algorithm Long-Term #8	Companion notebook for Mar. 3
Mar 13, 2006	An approximation for the behavior of the Gessel-Viennot algorithm in the weakly-interacting case. The result is not particularly interesting in itself, but is rather pretty simple and very good. I justify why this works.
Gessel-Viennot Algorithm Long-Term #9	Companion notebook for Mar. 13
Mar 14, 2006	An attempt at duplicating a polynomial expansion for the weakly-interacting case. It didn't really work out, since I don't have any very good point to take a series near.
Gessel-Viennot Algorithm Long-Term #10	Testing actual results from the Gessel-Viennot algorithm versus published weakly-interacting approximations.
Mar 21, 2006	I tried to find a valid form for the determinant in the N~P case, where I found some viable leads.
Gessel-Viennot Algorithm Long-Term #11	Tests relating to the earlier, less promising concepts from March 21.
Mar 28, 2006	I found a fascinating paper on Toeplitz matrices and their determinants: I found an integral form for the desired result, but unfortunately actually doing the integral is extraordinarily difficult.
Gessel-Viennot Algorithm Long-Term #12	Mathematica notebook companion to March 28.
Apr 4, 2006	Attempting to collapse the integral from last week
Gessel-Viennot Algorithm Long-Term #13	Companion Notebook for April 4
Apr 11, 2006	More exercises trying to take expansions of the integral by means of expansions. It seems to be cleverly avoiding all of my tricks.
Gessel-Viennot Algorithm Long-Term #14b	A mini-companion notebook trying a new approach which might have worked if it didn't turn out that the expansion contained worse mathematical functions than the original.
Apr 18, 2006	Finally, a nice-looking generating function that works. The problem now is that I can't sum the series!
Gessel-Viennot Algorithm Long-Term #15	Results verifying and supporting the work from April 18.
Apr 21, 2006	Agreement between the Gessel-Viennot derivation of the cubic correction and pulished results.
Gessel-Viennot Algorithm Long-Term #16	Supporting notebook for April 21.
Gessel-Viennot Algorithm Long-Term #17	This notebook is poorly documented, but is a few steps related to the method of steepest ascent that was a whim on how one might get more terms from the expansion found last week.
Poster Presentation Slides (Update 1)	A draft of my presentation for the first-year Poster Presentation day.
May 2, 2006	Another simple attempt at getting higher-order terms in the expansion: the integral of the logarithm of a sum is still difficult, and in this case would require finding difficult roots!