Despite the simplicity of their definition, self-avoiding walks (SAW) on the two-dimensional square lattice pose a number of open and possibly intractable problems. One such problem concerns the enumeration of the total number of SAW of length n emanating from the origin, while another related problem deals with the enumeration of all SAW, irrespective of length, but restricted to the finite subset {0,1,…,n}×{0,1,…,n} of the square lattice. At present no exact results, either in the form of a closed-form expression in terms of n or an explicit asymptotic estimate, is known for the enumerating function of either problem. Surprisingly in contrast, we shall in this paper obtain a closed-form expression for counting all SAW emanating from the origin but restricted to the finite square lattice strip {−a,…,0,…,b}×{0,1}, in terms of the non-negative integer parameters a and b. As shall be seen the argument used to establish this result will be both elementary and purely combinatorial in nature.
How to participate in this seminar:
1. Book your nearest ACE facility;
2. Notify Vera Roshchina at RMIT (maths.colloquia@rmit.edu.au) to notify you will be participating.
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