Chained Walks: Theory & Applications

• Chained Walk is a concept defining sequential, constrained motion along ordered states, applicable in probability, combinatorics, topology, and quantum systems.
• It encompasses models such as finite linear Markov chains with absorbing states, constrained lattice paths counted by Catalan numbers, and E-chains in uniform spaces.
• Recent extensions include memory-dependent ladder chains, tethered multi-agent walks, and quantum walks on chain-like graphs, offering diverse analytical and computational insights.

“Chained walk” does not designate a single universally fixed construction across the literature. In the classical probabilistic sense, it denotes a random walk on a finite linear Markov chain: a process whose state moves only to neighboring positions of a one-dimensional chain, often with absorbing endpoints. In adjacent literatures, the same expression or closely related ones also denote constrained lattice walks, finite sequences of uniformly close points in a uniform space, random walks with short-memory or tether constraints, and walks across hierarchically related simplices of different dimensions. The common structural feature is stepwise motion constrained by an underlying chain, boundary, hierarchy, or local coupling (Gil et al., 2017, Dershowitz, 2016, LaBuz, 2021).

1. Finite linear Markov chains

In its most direct probabilistic usage, a chained walk is a random walk on a finite chain of states $0,1,\dots,m$, with transitions only to neighboring states $i-1$ or $i+1$, together with possible absorbing endpoints. Formally, the process is a discrete-time Markov chain $X_0,X_1,X_2,\dots$ with transition probabilities

$P(X_{n+1}=j\mid X_n=i)=p_{ij},$

organized into a transition matrix $P=(p_{ij})$ with $\sum_j p_{ij}=1$. The defining Markov property is that the distribution of $X_{n+1}$ depends only on $X_n$, not on earlier history (Gil et al., 2017).

The elementary model in “Walks on chains” is the robot Cheburator on a short table. Four states encode falling off to the left, standing at the left edge, standing at the right edge, and falling off to the right. States $0$ and $i-1$0 are absorbing. If $i-1$1 and $i-1$2 denote the probabilities of eventual absorption at the right endpoint starting from the two table positions, then

$i-1$3

so $i-1$4 and $i-1$5. The corresponding left-absorption probabilities satisfy $i-1$6, $i-1$7, and $i-1$8. The survival probability after $i-1$9 steps is

$i+1$0

which tends to $i+1$1, and the expected absorption times satisfy $i+1$2 (Gil et al., 2017).

The long-table model is the standard simple symmetric random walk on $i+1$3 with absorbing boundaries at $i+1$4 and $i+1$5. If $i+1$6 is the probability of eventual absorption at $i+1$7, then

$i+1$8

hence $i+1$9. If the starting point is $X_0,X_1,X_2,\dots$0 steps from the left edge and $X_0,X_1,X_2,\dots$1 from the right, so $X_0,X_1,X_2,\dots$2, then

$X_0,X_1,X_2,\dots$3

For the expected absorption time $X_0,X_1,X_2,\dots$4,

$X_0,X_1,X_2,\dots$5

and the solution is

$X_0,X_1,X_2,\dots$6

In particular, $X_0,X_1,X_2,\dots$7, and in the symmetric case $X_0,X_1,X_2,\dots$8, $X_0,X_1,X_2,\dots$9. The same note interprets bacteria-color evolution as a walk on the chain $P(X_{n+1}=j\mid X_n=i)=p_{ij},$0 of green-bacteria counts, yielding

$P(X_{n+1}=j\mid X_n=i)=p_{ij},$1

and treats the drunk’s walk on positions $P(X_{n+1}=j\mid X_n=i)=p_{ij},$2 with reflecting behavior at the bar and absorption at home, for which the stated averages are $P(X_{n+1}=j\mid X_n=i)=p_{ij},$3 total blocks walked and $P(X_{n+1}=j\mid X_n=i)=p_{ij},$4 returns to the bar (Gil et al., 2017).

2. Constrained lattice paths and ordered chain games

A second usage places the walk on a higher-dimensional lattice but imposes a one-dimensional boundary constraint. In “Touchard’s Drunkard,” the walk is on $P(X_{n+1}=j\mid X_n=i)=p_{ij},$5 with step set

$P(X_{n+1}=j\mid X_n=i)=p_{ij},$6

starting at $P(X_{n+1}=j\mid X_n=i)=p_{ij},$7, constrained to remain in the half-plane $P(X_{n+1}=j\mid X_n=i)=p_{ij},$8, and required to end with equal numbers of $P(X_{n+1}=j\mid X_n=i)=p_{ij},$9- and $P=(p_{ij})$0-steps. Equivalently, every prefix satisfies $P=(p_{ij})$1, and the final walk returns to the promenade $P=(p_{ij})$2. The number $P=(p_{ij})$3 of such walks of length $P=(p_{ij})$4 satisfies

$P=(p_{ij})$5

where $P=(p_{ij})$6 is the Catalan number, and also

$P=(p_{ij})$7

The paper gives an explicit bijection from Dyck paths of length $P=(p_{ij})$8 by pairing steps according to

$P=(p_{ij})$9

and identifies Touchard walks with two-colored Motzkin paths. In this usage, the walk is “chained” to a half-plane by a ballot-type condition rather than to a finite segment of states (Dershowitz, 2016).

An adversarial variant appears in “Chain-making games in grid-like posets.” There the state space is a product of chains

$\sum_j p_{ij}=1$0

ordered coordinatewise, with levels defined by $\sum_j p_{ij}=1$1. In the unordered Maker–Breaker game on chains, if $\sum_j p_{ij}=1$2 is the maximum size of a chain in $\sum_j p_{ij}=1$3 and $\sum_j p_{ij}=1$4, then Maker can guarantee a chain of size

$\sum_j p_{ij}=1$5

and Breaker can prevent any larger one. The ordered Walker–Blocker variant requires Walker to move along the chain in order. On the wedge $\sum_j p_{ij}=1$6, the bottom $\sum_j p_{ij}=1$7 levels of the product of $\sum_j p_{ij}=1$8 arbitrarily long chains, Walker can guarantee a chain that hits all levels when $\sum_j p_{ij}=1$9. In dimension $X_{n+1}$0, the exact guarantee on $X_{n+1}$1 is only

$X_{n+1}$2

and $X_{n+1}$3 is asymptotically achievable in the product of two equal chains (Cranston et al., 2011).

These combinatorial models preserve the core idea of a chained walk—successive progress along ordered states—while shifting emphasis from stochastic transition rules to global path constraints or adversarial obstruction.

3. Topological and uniform-space formulations

In uniform-space theory, “chained” no longer refers primarily to a Markov chain but to a discretized path structure. Given a uniform space $X_{n+1}$4 and an entourage $X_{n+1}$5, an $X_{n+1}$6-chain is a finite sequence

$X_{n+1}$7

such that $X_{n+1}$8 for each $X_{n+1}$9. This is a discrete walk with uniformly controlled step size. Such chains correspond to simplicial edge-paths in the Rips complex $X_n$0, whose simplices are finite $X_n$1-bounded subsets (LaBuz, 2021).

A uniform space is chain connected if every pair of points can be joined by an $X_n$2-chain for every entourage $X_n$3. Two $X_n$4-chains with the same endpoints are $X_n$5-homotopic if one can be transformed into the other by inserting or deleting interior points while keeping endpoints fixed. A chain is $X_n$6-short if its $X_n$7-homotopy class equals that of the two-point chain $X_n$8. These notions lead to generalized paths, defined as coherent systems of chain-homotopy classes across all scales, and to the properties of uniform joinability and local uniform joinability (LaBuz, 2021).

Plaut’s notion of a weakly chained uniform space strengthens chain connectedness by requiring that for any entourage $X_n$9, some smaller entourage $0$0 has the property that every $0$1 admits arbitrarily fine chains $0$2 with

$0$3

For metrizable uniform spaces, weakly chained implies locally uniformly joinable. For metric continua, the note states the equivalences

$0$4

The “Texas circle” furnishes a path connected metric space that is uniformly joinable but not locally uniformly joinable, showing that the existence of generalized paths does not force local $0$5-shortness (LaBuz, 2021).

This topological usage preserves the stepwise, scale-controlled character of a walk while replacing probabilistic evolution by homotopy-sensitive connectivity.

4. Memory, coupling, and multi-level state spaces

Several later models retain the chained-walk intuition but alter the state description. In “Ladder Chains: A Variation of Random Walks,” the walk on $0$6 is chained to a short memory of recent Bernoulli outcomes. For integers $0$7, a ladder chain $0$8 is defined from i.i.d. Bernoulli variables $0$9 by step increments

$i-1$00

Thus $i-1$01 is not Markov in its own state variable alone but is an $i-1$02-th order Markov chain. For $i-1$03, the step sizes are $i-1$04, and the drift is

$i-1$05

The critical value is

$i-1$06

and the paper proves recurrence of $i-1$07 (Zhang et al., 2018).

A different form of chaining appears in “Tethered single-legged molecular spiders on independent 1D tracks.” There the basic components are one-legged walkers on parallel one-dimensional tracks, each with hopping rate $i-1$08 on product and $i-1$09 on substrate, connected by a leash imposing the kinematic constraint that no two spiders can be more than a certain distance apart. A single one-legged walker does not exhibit directional, superdiffusive motion, but a team of one-legged walkers connected by a flexible tether does enjoy a superdiffusive transient. In the $i-1$10 analysis for two one-legged spiders with leash length $i-1$11, the expected number of team steps per boundary period is

$i-1$12

The paper further states that one-legged walker teams exhibit a greater expected number of steps per boundary period and diffuse more quickly through the product sea than two-legged walkers, leading to longer periods of superdiffusion (1909.01872).

The simplicial-complex model “Random Walks Across Dimensions: Exploring Simplicial Complexes” generalizes the state space itself. The walker moves not within a fixed dimension but across dimensions,

$i-1$13

through a block-structured stochastic matrix $i-1$14 acting on

$i-1$15

The stationary probability of a $i-1$16-simplex $i-1$17 is proportional to its generalized degree

$i-1$18

thereby inducing a ranking of nodes, edges, triangles, and higher simplices. The paper then augments the walk with stochastic teleportation,

$i-1$19

and studies optimal search strategies via mean first passage times (Febbe et al., 22 Jan 2026).

Across these models, “chained” refers not merely to a line of states but to dependence on recent outcomes, to kinematic coupling among walkers, or to hierarchical adjacency between dimensions.

5. Sampled chains, cover times, and induced processes

A further extension studies Markov chains observed only at selected events. For a Markov chain $i-1$20 on a Polish space and a set $i-1$21, “Stationary entrance chains and applications to random walks” defines the entrance times

$i-1$22

the entrance chain

$i-1$23

and the exit chain

$i-1$24

These sampled processes are again time-homogeneous Markov chains. If $i-1$25 has invariant measure $i-1$26, the induced entrance and exit measures are

$i-1$27

In one dimension, for an oscillating random walk, the overshoot chain at zero crossings has invariant measure

$i-1$28

and this ergodic structure yields a central limit theorem for the number of level crossings $i-1$29: $i-1$30 Here the chain is not spatially linear in the original dynamics but is induced by repeated entrances and exits across a boundary (Mijatovic et al., 2024).

The analytic study of such walks also uses chaining in a different sense. In “Cover times and generic chaining,” for an irreducible positive recurrent Markov chain on state space $i-1$31, the commute-time metric is

$i-1$32

and the cover time of a finite set $i-1$33 is controlled by Talagrand’s generic-chaining functional. The paper proves

$i-1$34

for every finite $i-1$35, and under reversibility,

$i-1$36

It also quotes the Ding–Lee–Peres theorem that for finite reversible chains,

$i-1$37

Here “chaining” is not the walk itself but the multiscale method used to quantify how long a walk needs to cover its state space (Lehec, 2012).

6. Quantum and graph-theoretic chain-like walks

In continuous-time quantum-walk theory, “chained walk” can refer to transport on chain-like graphs built by repeating small graph units. “Enhanced quantum transport in chiral quantum walks” studies continuous-time quantum walks on long chains formed by repeating units such as $i-1$38, $i-1$39, and $i-1$40, with transport probability

$i-1$41

The Hamiltonian allows complex edge phases,

$i-1$42

so loops support gauge-invariant chiral phases. The paper identifies three candidate chain-unit structures with optimal performance. One, the $i-1$43 chain, can be reduced to a weighted line with central couplings $i-1$44. The other two, the $i-1$45 chain and the $i-1$46 chain, are described as truly chiral quantum walks, with enhanced transport probability over long chain structures. Their transport is analyzed through a first-maxima criterion in the time window

$i-1$47

and through Krylov reduction to effective weighted-line models when possible (Annoni et al., 2023).

This quantum usage retains the linear repetition implicit in a chain while replacing stochastic transition probabilities by unitary evolution and interference. A plausible implication is that the phrase “chained walk” has broadened from a walk constrained to a linear set of states to a family of models whose dynamics are organized by sequential, locally coupled structure.

Taken together, these literatures show that “Chained Walk” is best understood as a family of formally distinct constructions unified by constrained sequential motion. In elementary probability it is a random walk on a finite linear Markov chain with formulas such as $i-1$48 and $i-1$49; in enumerative combinatorics it is a boundary-constrained lattice path counted by Catalan numbers; in topology it is an $i-1$50-chain in a uniform space; in modern stochastic models it may be chained by memory, tethering, or higher-order incidence; and in quantum transport it may be a walk on a linearly repeated graph with chiral phases. The persistence of the term across these settings reflects a shared abstraction: progression by local steps along a structure that restricts how motion can continue.