Self-Osculating Walks (SOWs)

• Self-osculating walks are lattice paths that permit controlled vertex touchings (osculations) without crossing, generalizing self-avoiding walks.
• Rigorous enumeration uses automata and transfer-matrix methods to derive connective constant bounds, e.g., μ_SOW_□ ≤ 2.73911 on the square lattice.
• Their analysis is lattice-dependent, with distinct formulations on square, triangular, and hexagonal lattices, offering insights into higher-dimensional osculating structures.

Searching arXiv for the specified papers to ground the article in current sources. {"query":"id:(Kim et al., 4 Sep 2025) OR id:(Owczarek et al., 23 Jul 2025)", "max_results": 5} Self-osculating walks (SOWs) are restricted lattice walks that generalize self-avoiding walks (SAWs) by allowing strands to touch, or “kiss,” at a vertex while still forbidding crossings; they are analogous to osculating polygons except that they are not required to be closed (Kim et al., 4 Sep 2025). In the recent restricted-walk framework, they are treated as geometric vertex models specified by admissible local configurations rather than by a simple no-repeated-vertex rule, and they occupy an intermediate position between SAWs and edge-avoiding walks in the standard inclusion hierarchy (Kim et al., 4 Sep 2025). Closely related but distinct exact results concern directed systems of mutually osculating walks with asymmetric interactions; these models illuminate osculation without edge sharing, but they are not single-chain SOWs (Owczarek et al., 23 Jul 2025).

1. Definition and admissibility

The defining modification from SAWs to SOWs is local and geometric. A SAW on a graph is a sequence of successively adjacent vertices in which no vertex is visited twice. For SOWs, by contrast, certain multiple incidences at a vertex are allowed, provided that the local geometry is resolved as an osculation rather than a crossing. The central rule is therefore: crossings are forbidden, but a would-be crossing can be replaced by one of two local osculating resolutions in which strands only touch at the vertex (Kim et al., 4 Sep 2025).

This distinction matters because SOWs are not defined by a simple graph-theoretic condition analogous to “no vertex visited twice.” The allowed local configurations are specified through bulk and boundary vertex configurations. Repeated edges are also excluded: the hierarchy places SOWs strictly inside edge-avoiding walks,

$${DP}_n \subset {NAW}_n \subset {SAW}_n \subseteq {ODW}_n \subseteq {SOW}_n \subset {EAW}_n \subset {NRW}_n \subset {RW}_n,$$

and correspondingly

$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$

Thus SOWs relax self-avoidance at vertices in a controlled manner, but they remain substantially more constrained than generic edge-avoiding or non-reversing walks (Kim et al., 4 Sep 2025).

A common source of confusion is to treat SOWs as arbitrary walks with vertex revisits. The vertex-model formulation rules this out: admissibility depends on the local pairing structure at a multiply incident vertex, not merely on whether a vertex has been encountered before. This suggests that SOWs are best understood as geometric objects with non-crossing local reconnections rather than as a minor perturbation of SAW vertex counting.

2. Lattice-dependent formulations

The local realization of SOWs is strongly lattice dependent. On the square lattice, the model is generated from two local configurations: the straight-through occupied vertex and the osculating “double-corner” vertex; all other allowed bulk and boundary configurations are obtained by rotation, reflection, and truncation. On the triangular lattice, the admissible set is larger: one first specifies fully occupied admissible local patterns up to rotations and reflections, and then takes all subsets of those maximal patterns as allowed vertices. On the hexagonal lattice, the degree is only $3$, so crossing-like four-leg encounters cannot occur (Kim et al., 4 Sep 2025).

Lattice	Local definition	Relation to adjacent classes
Square	Generated by straight-through and osculating “double-corner” vertices, plus rotations, reflections, and truncations	${ODW}_\square = {SOW}_\square$
Triangular	Allowed vertices are subsets of fully occupied admissible generators up to rotations and reflections	${ODW}_\triangle \subset {SOW}_\triangle$
Hexagonal	No crossing-like four-leg local encounter is possible	$\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$

The distinction between osculating domain wall walks (ODWs) and SOWs is especially significant on the triangular lattice. The ODW construction is generated from Boolean face variables. On the square lattice it reproduces all SOW local configurations, but on the triangular lattice it does not: the crossing-to-osculation rule permits local SOW configurations that are not domain-wall realizable, hence ${ODW}_\triangle \subset {SOW}_\triangle$ (Kim et al., 4 Sep 2025).

This lattice dependence has structural consequences. On the square lattice, the equality ${ODW}_\square = {SOW}_\square$ makes domain-wall techniques directly relevant to SOW enumeration. On the triangular lattice, the strict inclusion shows that SOWs are genuinely broader than the domain-wall subclass.

3. Enumeration, connective constants, and existence

For SOWs, the counting sequence is denoted $c_n$, where the walk length is the number of visited vertices excluding the starting vertex $v_0$. The finite-$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$0 growth estimator is

$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$1

and the connective constant is

$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$2

For walks, including SOWs, existence follows from subadditivity: any valid walk of length $$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$3 decomposes into initial and terminal segments, whereas not every pair of valid length-$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$4 and length-$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$5 walks can be concatenated into a valid length-$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$6 walk. Hence

$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$7

so $$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$8 is subadditive and Fekete’s lemma implies existence of the limit (Kim et al., 4 Sep 2025).

The same argument yields the standard inequality

$$\mu^{DP} \leq \mu^{NAW} \leq \mu^{SAW} \leq \mu^{ODW} \leq \mu^{SOW} \leq \mu^{EAW} \leq \mu^{NRW} < \mu^{RW}.$$9

which is why finite enumerations immediately provide rigorous upper bounds. For square-lattice SOWs, the paper lists the first 18 values of $3$0, including

$3$1

From the $3$2 count one obtains the direct upper bound

$3$3

Because SOWs are supersets of SAWs, known SAW lower bounds also transfer. On the square lattice,

$3$4

so in particular

$3$5

(Kim et al., 4 Sep 2025).

These facts place SOWs within the standard asymptotic framework of lattice-walk enumeration. The novelty lies not in the existence of $3$6, but in the difficulty of obtaining sharp bounds once vertex osculation is permitted.

4. Automata method and rigorous upper bounds

The principal rigorous bounds for SOWs are obtained by adapting the Pönitz–Tittmann automata method. For a restricted-walk rule $3$7, one defines an indicator $3$8 for path admissibility. A loop of size $3$9 is then a minimal forbidden path of length ${ODW}_\square = {SOW}_\square$0: a disallowed path ${ODW}_\square = {SOW}_\square$1 with ${ODW}_\square = {SOW}_\square$2 such that every subpath

${ODW}_\square = {SOW}_\square$3

is allowed. The method counts walks that avoid all loops up to size ${ODW}_\square = {SOW}_\square$4, and because every SOW avoids those loops, the resulting counts dominate the true SOW counts (Kim et al., 4 Sep 2025).

If ${ODW}_\square = {SOW}_\square$5 denotes the number of length-${ODW}_\square = {SOW}_\square$6 walks with no loops of size ${ODW}_\square = {SOW}_\square$7, then

${ODW}_\square = {SOW}_\square$8

where ${ODW}_\square = {SOW}_\square$9 is the lattice coordination number, ${ODW}_\triangle \subset {SOW}_\triangle$0 is the initial one-step path, ${ODW}_\triangle \subset {SOW}_\triangle$1 is the all-ones vector, and ${ODW}_\triangle \subset {SOW}_\triangle$2 is the transfer matrix built from loop-avoiding suffix states. The state space ${ODW}_\triangle \subset {SOW}_\triangle$3 keeps only the last ${ODW}_\triangle \subset {SOW}_\triangle$4 steps, because once a path is known to avoid short forbidden loops, only the newest suffix can create a new loop of size ${ODW}_\triangle \subset {SOW}_\triangle$5. The rigorous upper bound on the connective constant is the spectral radius, or largest eigenvalue, of ${ODW}_\triangle \subset {SOW}_\triangle$6 (Kim et al., 4 Sep 2025).

On the square lattice, the minimal forbidden loops up to size ${ODW}_\triangle \subset {SOW}_\triangle$7 consist of the backtracking loop of length ${ODW}_\triangle \subset {SOW}_\triangle$8 and three distinct loops of length ${ODW}_\triangle \subset {SOW}_\triangle$9. The corresponding automaton yields a $\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$0 transfer matrix with largest eigenvalue

$\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$1

Increasing the maximum accounted loop size improves the upper bound monotonically: $\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$2 for maximum loop sizes

$\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$3

respectively. This yields the theorem

$\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$4

On the triangular lattice, the loop structure is richer. Excluding the trivial back-and-forth loop of length $\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$5, loops occur at all lengths from $\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$6 onward. The trivial $\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$7 bound is

$\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$8

and extending the accounted loop size to $\mathrm{SAW}_{\hexagon} = \mathrm{SOW}_{\hexagon}$9 gives the sequence

${ODW}_\triangle \subset {SOW}_\triangle$0

for maximum loop sizes

${ODW}_\triangle \subset {SOW}_\triangle$1

hence the theorem

${ODW}_\triangle \subset {SOW}_\triangle$2

(Kim et al., 4 Sep 2025).

These results are rigorous upper bounds rather than exact connective constants. Their importance is methodological: the local-configuration definition of SOWs is compatible with finite-state transfer-matrix truncations once forbidden short loops are organized appropriately.

5. Exact directed osculating analogues

A distinct but closely related development is the exact solution of a model of three directed osculating walks in a star configuration on the square lattice with asymmetric pair-contact Boltzmann weights ${ODW}_\triangle \subset {SOW}_\triangle$3 and ${ODW}_\triangle \subset {SOW}_\triangle$4 (Owczarek et al., 23 Jul 2025). Each walk consists of ${ODW}_\triangle \subset {SOW}_\triangle$5 directed steps, each step being either

${ODW}_\triangle \subset {SOW}_\triangle$6

the square lattice is rotated by ${ODW}_\triangle \subset {SOW}_\triangle$7, the walks start at ${ODW}_\triangle \subset {SOW}_\triangle$8 at heights

${ODW}_\triangle \subset {SOW}_\triangle$9

and they are non-crossing and ordered as top, middle, and bottom throughout. In this model, “osculating” means that walks may share vertices, may not share edges, and may not cross; triple contacts do not occur (Owczarek et al., 23 Jul 2025).

The model is not, strictly speaking, a SOW: it is a system of three mutually osculating directed walks rather than a single walk with self-contacts. Its relevance to SOWs lies in the shared local geometric feature—contact at vertices without edge sharing—and in the exact thermodynamic analysis of osculating contacts. With ${ODW}_\square = {SOW}_\square$0 the number of top–middle shared sites and ${ODW}_\square = {SOW}_\square$1 the number of middle–bottom shared sites, the partition sum is

${ODW}_\square = {SOW}_\square$2

and the star generating function is

${ODW}_\square = {SOW}_\square$3

The exact solution shows that the star generating function in the asymmetric case is algebraic of degree ${ODW}_\square = {SOW}_\square$4: if ${ODW}_\square = {SOW}_\square$5, then

${ODW}_\square = {SOW}_\square$6

with polynomial coefficients ${ODW}_\square = {SOW}_\square$7. The phase diagram has four phases, defined by the contact densities ${ODW}_\square = {SOW}_\square$8: free, top zipped, bottom zipped, and fully zipped. The corresponding entropic exponents are

• free: ${ODW}_\square = {SOW}_\square$9,
• top-zipped: $c_n$0,
• bottom-zipped: $c_n$1,
• fully zipped: $c_n$2,

with $c_n$3 on the free-to-partially-zipped boundaries and $c_n$4 at the multicritical point $c_n$5. The paper further states that all phase transitions are second-order, with

$c_n$6

For SOW-oriented research, this model functions as an exact benchmark for osculation without edge sharing. It should not be cited as a canonical single-chain SOW, but it is directly informative about how asymmetric osculating-contact weights can generate free, partially zipped, and fully zipped regimes. This suggests that exact solvability in osculating systems depends strongly on directedness, non-crossing order, and star geometry.

6. Generalizations and conceptual scope

SOWs are presented as the $c_n$7 member of a broader osculating-geometry program. In the same framework, one defines self-osculating surfaces (SOSs) with signature $c_n$8 and self-osculating manifolds (SOMs) with signature $c_n$9; the corresponding constructions are related to $v_0$0-XDs and to self-avoiding $v_0$1-manifolds (SAMs) (Kim et al., 4 Sep 2025). By adapting the concatenation procedure of van Rensburg and Whittington, 1989, the paper proves that their growth constants exist and gives an explicit form for their upper and lower bounds. The upper bounds can be improved by adapting the “twig” method, originally developed for polyominoes (Kim et al., 4 Sep 2025).

Within this broader setting, several distinctions are essential. First, SOWs are the natural open-walk analogue of osculating polygons, not a minor restatement of SAWs. Second, the equality $v_0$2 is lattice specific; on the triangular lattice, $v_0$3, so domain-wall realizability does not exhaust osculating admissibility (Kim et al., 4 Sep 2025). Third, SOWs should not be conflated with friendly-walk models: in the directed three-walk literature, friendly walks may share both vertices and edges, whereas osculating walks may share vertices only and may not share edges (Owczarek et al., 23 Jul 2025).

A plausible implication is that “osculation” is not a single combinatorial notion but a family of related local-contact constraints whose enumerative and thermodynamic consequences depend sharply on lattice degree, local pairing rules, and global geometry. In that sense, SOWs occupy a precise intermediate position: they retain the non-crossing discipline of self-avoidance while admitting vertex contact structures rich enough to require dedicated local-state and transfer-matrix machinery.