Hamiltonian and exclusion statistics approach to discrete forward-moving paths

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an alternative, simpler form for its logarithm that makes its polynomial structure explicit.

Figure 1

A typical path [solid (purple) meander] for $k=4$, starting at $m=1$ and ending at $n=2$, of length $l=13$ steps. The area under it is 27.5 plaquettes and subtracting the area of the triangular “sierra” at the bottom $l/2=6.5$ we obtain $A=21$. Relevant to the conventions after Sec. 4a, its length can also be expressed as $l=6.5$ double steps and $A=21/2=10.5$ “diamonds” formed by the dashed lines (there is a half-diamond at the last step of the path).

Figure 2

A geometric interpretation of recursion relation (3.21) as a “first passage” equation. Any excursion (one of length 24 pictured above) can be decomposed into a first-passage path returning to height $j=0$ for the first time [first (blue and orange) part of path] and the remaining (purple) arbitrary excursion. For paths of length at least two, the first-passage path has a first and a last step [lower (blue) links]. The remaining upper (orange) part never dips below $j=1$ and can be interpreted as an excursion, but with a ceiling reduced by 1 and an area increased by its length (shaded plaquettes). Summing over all paths gives the full generating function $G_k(Z,Q)$ as the product of itself [second (purple) excursion] times $G_{k-1}(ZQ,Q)$ [first (orange) excursion], the shift $Z\to ZQ$ accounting for the increase in the area, times $Z^2$ contributed by the two lower (blue) links. The trivial path of zero length and area cannot be decomposed and contributes the term 1 in (3.21). Relations (3.18) also admit a similar first-passage interpretation.

Figure 3

A “tiled roof” path of maximal area for $k=4$, starting at $m=1$ and ending at $n=2$, of length $l=2a+n-m=13$, so for $a=6$, zigzagging $a-k+n=4$ times at the ceiling. The total area in diamonds is given by the second term in (5.16) and the $lnq$ coefficient in (5.14) as $(k-n-1)(2a-k+n)/2+an+(n-m)(n+m-1)/4=17.5$, agreeing with the number of diamonds in the figure.