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Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height

Jay Pantone, Alexander R. Klotz, Everett Sullivan

TL;DR

The paper develops an exact, solvable framework for growing self-avoiding walks on half-infinite grid strips by constructing a finite-state machine that encapsulates GSAW building blocks as frames. Using a transfer-matrix approach on the derived directed graph, the authors obtain rational generating functions for trapping length and displacement across heights up to at least 6 (and for Greek key tours up to height 8), enabling precise mean lengths and displacements and revealing connective constants that match known confined-walk values. The methodology extends to two probabilistic models—uniform and energetic—by incorporating edge probabilities either directly or via a parameter C, yielding exact generating functions and moment calculations, and it is complemented by Monte Carlo simulations that connect confined-strip results to the unconfined, quarter-infinite plane. The work provides a comprehensive, exactly-solvable treatment of GSAWs in confined geometries, yields asymptotic growth rates, and resolves conjectures about Greek key tours, with implications for polymer modeling in nanochannels and related systems. Overall, the combination of FSM construction, transfer-matrix analysis, and probabilistic extensions offers a powerful toolkit for enumerating and analyzing trapped growth processes on lattices.

Abstract

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.

Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height

TL;DR

The paper develops an exact, solvable framework for growing self-avoiding walks on half-infinite grid strips by constructing a finite-state machine that encapsulates GSAW building blocks as frames. Using a transfer-matrix approach on the derived directed graph, the authors obtain rational generating functions for trapping length and displacement across heights up to at least 6 (and for Greek key tours up to height 8), enabling precise mean lengths and displacements and revealing connective constants that match known confined-walk values. The methodology extends to two probabilistic models—uniform and energetic—by incorporating edge probabilities either directly or via a parameter C, yielding exact generating functions and moment calculations, and it is complemented by Monte Carlo simulations that connect confined-strip results to the unconfined, quarter-infinite plane. The work provides a comprehensive, exactly-solvable treatment of GSAWs in confined geometries, yields asymptotic growth rates, and resolves conjectures about Greek key tours, with implications for polymer modeling in nanochannels and related systems. Overall, the combination of FSM construction, transfer-matrix analysis, and probabilistic extensions offers a powerful toolkit for enumerating and analyzing trapped growth processes on lattices.

Abstract

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.
Paper Structure (24 sections, 5 theorems, 71 equations, 20 figures, 4 tables)

This paper contains 24 sections, 5 theorems, 71 equations, 20 figures, 4 tables.

Table of Contents

  1. Growing Self-Avoiding Walks
  2. Prior Literature and Summary of Main Results
  3. The Non-Probabilistic Finite State Machine Construction
  4. The Frames of a GSAW
  5. Finite State Machines
  6. Construction of the Directed Graph
  7. A Bijection Between Walks and GSAWs
  8. Non-Probabilistic Results
  9. Height 2
  10. Height 3
  11. Height 4
  12. Height 5
  13. Height 6
  14. Height 7
  15. The Probabilistic Finite State Machine Constructions
  16. ...and 9 more sections

Key Result

Theorem 3.1

Walks in $D_h$ that start at the start vertex, end at an accepting vertex, and have weight $x^ny^k$ (denoted $\mathcal{W}_{n,k}$) are in bijection with GSAWs with $n$ edges and displacement $k$ (denoted $\mathcal{S}_{n,k}$). As a consequence, we can find the generating function for GSAWs on $\mathca

Figures (20)

  • Figure 1: On the left, the finite grid graph $\mathcal{G}_{\{0, \ldots, 3\} \times \{0, \ldots, 2\}}$. On the right, the half-infinite grid graph $\mathcal{G}_4$.
  • Figure 2: A GSAW on $\mathcal{G}_3$ that starts at $(0,2)$ and ends trapped at $(1,2)$. The probability of each edge is shown. The probability that this GSAW occurs on $\mathcal{G}_3$ is the product of the edge probabilities, $1/576$.
  • Figure 3: A GSAW on $\mathcal{G}_5$ with length $65$ and displacement $14$.
  • Figure 4: The leftmost four frames of the GSAW in Figure \ref{['figure:frame-ex-1']}.
  • Figure 5: On the left, four frames such that each consecutive pair is compatible. On the right, the non-GSAW graph you get when you join them all together.
  • ...and 15 more figures

Theorems & Definitions (9)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 7.1