Exact transfer-matrix enumeration and critical behaviour of self-avoiding walks across finite strips

1994 ◽  
Vol 27 (12) ◽  
pp. 4055-4067 ◽  
Author(s):  
M T Batchelor ◽  
C M Yung
Keyword(s):  
Transfer Matrix ◽  
Critical Behaviour ◽  
Self Avoiding Walks

scholarly journals SPECTRUM OF THE LOOP TRANSFER MATRIX ON FINITE LATTICES

2001 ◽  
Vol 16 (16) ◽  
pp. 1069-1077 ◽  
Author(s):  
GEORGIOS DASKALAKIS ◽  
GEORGE K. SAVVIDY
Keyword(s):  
Transfer Matrix ◽  
Phase Structure ◽  
Extrinsic Curvature ◽  
Critical Behaviour ◽  
Equivalent Representation ◽  
Random Surfaces ◽  
Finite Dimensional ◽  
Curvature Term ◽  
Finite Lattices ◽  
Euclidean Lattice

We consider a model of random surfaces with extrinsic curvature term embedded into 3-D Euclidean lattice Z3. On a 3-D Euclidean lattice it has an equivalent representation in terms of the transfer matrix K(Qi, Qf), which describes the propagation of the loops Q. We study the spectrum of the transfer matrix K(Qi, Qf) on finite-dimensional lattices. The renormalisation group technique is used to investigate the phase structure of the model and its critical behaviour.

Critical behaviour of directed self-avoiding walks

1983 ◽  
Vol 16 (4) ◽  
pp. L113-L116 ◽  
Author(s):  
B K Chakrabarti ◽  
S S Manna
Keyword(s):  
Critical Behaviour ◽  
Self Avoiding Walks

The Umbral Transfer-Matrix Method. IV. Counting Self-Avoiding Polygons and Walks

10.37236/1572 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Doron Zeilberger
Keyword(s):  
Transfer Matrix ◽  
Cross Section ◽  
Polynomial Time ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vertical Cross Section ◽  
Trivial Case ◽  
Exponential Time ◽  
Self Avoiding Walks ◽  
Special Case

This is the fourth installment of the five-part saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota's seminal notion of the umbra. In this article we describe the Maple packages USAP, USAW, and MAYLIS. USAP automatically constructs, for any specific $r$, an Umbral Scheme for enumerating, according to perimeter, the number of self-avoiding polygons with $\leq 2r$ horizontal edges per vertical cross-section. The much more complicated USAW does the analogous thing for self-avoiding walks. Such Umbral Schemes enable counting these classes of self-avoiding polygons and walks in polynomial time as opposed to the exponential time that is required by naive counting. Finally MAYLIS is targeted to the special case of enumerating classes of saps with at most two horizontal edges per vertical cross-section (equivalently column-convex polyominoes by perimeter), and related classes. In this computationally trivial case we can actually automatically solve the equations that were automatically generated by USAP. As an example, we give the first fully computer-generated proof of the celebrated Delest-Viennot result that the number of convex polyominoes with perimeter $2n+8$ equals $(2n+11)4^n-4(2n+1)!/n!^2$.

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