Locally Activated Random Walks (LARWs)

• Locally Activated Random Walks are stochastic processes where local activation rules alter jump rates or persistence, making the dynamics non-Markovian with aging effects.
• They exhibit diverse behaviors such as anomalous diffusion, ballistic escape, and trapping on directed circuits, depending on activation functions and graph geometry.
• Analytical methods like renewal theory, Laplace transforms, and martingale techniques reveal critical transitions and self-organized criticality in both single-particle and interacting-particle frameworks.

Locally Activated Random Walks (LARWs) are stochastic processes in which motion is modified by local information generated by the trajectory or by local activation–deactivation rules. In the literature represented here, the label is used for distinct but structurally related families: single-particle models in which a walker’s jump rate, diffusivity, or persistence changes after visits to designated activation sites; reinforced walks in which transition probabilities depend on local times or directed crossing numbers; and interacting particle systems of Activated Random Walk type, where sleeping particles are reactivated when an active particle lands on their site. The unifying feature is that the dynamics is governed by a local memory field or local configuration, so that the position process alone is typically non-Markovian, whereas a suitably enlarged state space restores Markovian structure (Brémont et al., 2023, Erhard et al., 2019, Asselah et al., 2022).

1. Conceptual scope and defining mechanisms

A canonical single-particle formulation places a walker on $\mathbb{Z}^d$ with one activation site at the origin and an activation variable

$a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$

the cumulative time spent at the activation site. The jump rate is then $1/\tau(a)$, so local history modifies future mobility through a scalar internal state. In active variants, a persistence parameter $\varepsilon(a)$ is also activation-dependent. This yields a class of strongly non-Markovian, aging processes in which local occupation feeds back into transport coefficients (Brémont et al., 2023).

A second formulation uses local time directly in the transition rule. In the directed edge reinforced model called the Ant RW, the state variable is the net directed crossing number

$$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$

and the weight of the directed edge $(x,y)$ at time $n$ is

$a_n(x,y)=\exp\{\beta c_n(x,y)\}.$

The next step from the current vertex depends only on the neighboring directed edges and these local crossing numbers, so the reinforcement is local in space but cumulative in time (Erhard et al., 2019).

A third formulation is the interacting-particle Activated Random Walk (ARW). Here particles are active or sleeping; active particles perform simple random walks, an isolated active particle falls asleep at rate $\lambda$, and a sleeping particle is instantaneously reactivated when an active particle arrives at its site. Activation and deactivation are therefore purely local operations on the site configuration (Basu et al., 2017, Asselah et al., 2022).

This diversity is important because “local activation” does not specify a unique model class. It specifies a mechanism: dynamical parameters or state transitions are altered by local occupation, local time, or local contact events. The resulting phenomenology includes aging, anomalous diffusion, ballisticity, trapping on small structures, fixation, and self-organized criticality.

2. Single-particle activation at localized sites

For lattice LARWs with one activation site, the basic object is the joint law $P(x,a,t)$ of position and activation. The general framework developed for passive and persistent walks expresses the Laplace transform of the joint law in terms of the non-activated propagator: $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$0 This reduces the activated problem to renewal theory and generating functions for the underlying non-activated walk. In one dimension, local activation can induce anomalous diffusion; in two and three dimensions the typical displacement remains diffusive, $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$1, but the long-time distributions remain strongly non-Gaussian. For persistent LARWs there is an exact asymptotic reduction to an effective non-persistent model with renormalized waiting time

$a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$2

A sharp trapping criterion is also available: $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$3 For recurrent underlying walks in $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$4, trapping in this regime is complete; in transient dimensions the trapped fraction is strictly smaller than one (Brémont et al., 2023).

The continuous-space Brownian counterpart replaces occupation time by local time at the origin. In the rigorous formulation, the pair $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$5 satisfies

$a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$6

with local time $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$7 at $a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$8. The generator is

$a(t)=\int_0^t \mathbf{1}\big(x(t')=0\big)\,dt',$9

and the forward equation is

$1/\tau(a)$0

For $1/\tau(a)$1, there is a dynamical transition: if $1/\tau(a)$2, the process reaches the absorbing static state $1/\tau(a)$3 in finite time almost surely; if $1/\tau(a)$4, $1/\tau(a)$5 without complete trapping; if $1/\tau(a)$6, the process is recurrent. For positive $1/\tau(a)$7, the diffusivity can diverge, and the position distribution becomes non-Gaussian and multi-peaked. The earlier formulation of locally activated diffusion already identified the same phenomena: non-Gaussianity, off-center maxima under acceleration, and a trapping transition under sufficiently strong local deceleration (Meunier et al., 2016, Bénichou et al., 2011).

These single-particle models establish a central point: even one localized activation site can destroy Gaussianity, generate explicit aging, and produce either effective repulsion from the activated site or irreversible trapping there.

3. Directed reinforcement and circuit localization

The Ant RW provides a paradigmatic discrete-time LARW driven by directed local time. On a connected locally finite undirected graph $1/\tau(a)$8, the transition rule from the current vertex $1/\tau(a)$9 is

$\varepsilon(a)$0

so the walk is exponentially attracted to directions with positive net crossing number and exponentially repelled from the reverse direction. Because $\varepsilon(a)$1, reinforcement is non-monotone at the undirected level: backtracking cancels accumulated preference rather than increasing it (Erhard et al., 2019).

The long-time behavior depends sharply on graph geometry. On any finite connected non-tree graph, and on $\varepsilon(a)$2 for $\varepsilon(a)$3, the walk almost surely becomes trapped in some directed circuit that it follows forever. On $\varepsilon(a)$4, by contrast, the process becomes Markovian and satisfies an explicit law of large numbers: $\varepsilon(a)$5 Thus one dimension yields ballistic escape with random sign, whereas higher dimensions and cyclic finite graphs yield localization on periodic directed orbits (Erhard et al., 2019).

This mechanism differs from classical reinforced random walks. In the broad reinforced-walk literature, edge reinforcement under strong enough weights can localize on a single undirected edge, while vertex reinforcement on $\varepsilon(a)$6 with linear weights localizes almost surely on exactly five sites and, for certain superlinear weights, on two sites. The techniques there include martingales tracking asymmetry of local times and the Rubin time-lines construction. A plausible implication is that directed, net-crossing reinforcement changes the stable trap geometry: a two-vertex back-and-forth pattern is unstable because reverse traversals cancel the reinforcement, whereas a directed circuit preserves it (Tarrès, 2011, Erhard et al., 2019).

4. Activated Random Walks as interacting-particle LARWs

In ARW, local activation is no longer an internal state of one walker but a many-body interaction. On $\varepsilon(a)$7, each active particle performs a continuous-time simple random walk, sleeps at rate $\varepsilon(a)$8 when alone, and any sleeping particle is reactivated instantly when an active particle shares its site. This model has a fixation–activity phase transition governed by a critical density $\varepsilon(a)$9. A general theorem shows that

$$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$0

for every $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$1 and every $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$2, and gives explicit upper bounds in the small- and large-$$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$3 regimes. In two dimensions, for example, the bounds take the forms $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$4 for small $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$5 and $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$6 for large $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$7; in $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$8, the corresponding scales are $$c_n(x,y)=\sum_{k=0}^{n-1}\Big(\mathbf{1}\{(X_k,X_{k+1})=(x,y)\}-\mathbf{1}\{(X_k,X_{k+1})=(y,x)\}\Big),$$9 and $(x,y)$0 (Asselah et al., 2022).

Finite-volume ARW exhibits the same critical structure through time scales. On the cycle $(x,y)$1, with Bernoulli($(x,y)$2) initial active particles, the fixation time is at most $(x,y)$3 with high probability when $(x,y)$4, whereas for every fixed $(x,y)$5 there exists $(x,y)$6 such that for $(x,y)$7 the fixation time is at least $(x,y)$8 with probability $(x,y)$9. This finite-cycle dichotomy mirrors the infinite-volume fixation/non-fixation transition (Basu et al., 2017).

The village model of ARW (VARW) inserts a mean-field scale inside each site: each vertex of a finite underlying graph is replaced by $n$0 houses, jumps between villages follow a strictly sub-stochastic irreducible kernel $n$1, and sleep rates may depend on the village. Under the subcriticality condition $n$2, the scaled odometer $n$3 and final sleepy density $n$4 converge in probability to deterministic limits $n$5 characterized by

$n$6

$n$7

The fixed-point map governing $n$8 is contractive in a weighted norm associated with the Perron–Frobenius eigenvalue of $n$9, which supplies uniqueness and a law of large numbers for the stable configuration (Ráth et al., 7 May 2026).

5. Critical density, local density, and finite-volume relaxation

In one dimension, recent work has unified several critical-density notions for ARW. Let $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$0 be the number of sleeping particles in the stationary distribution of the driven–dissipative ARW chain on $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$1, obtained by repeatedly adding one active particle at a uniformly chosen site and stabilizing with boundary killing. A new proof of superadditivity shows that $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$2 stochastically dominates the sum of independent copies of $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$3 and $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$4. From this one gets convergence in probability

$a_n(x,y)=\exp\{\beta c_n(x,y)\}.$5

together with exponential lower-tail bounds, and one can deduce both the hockey-stick law

$a_n(x,y)=\exp\{\beta c_n(x,y)\}.$6

for the driven–dissipative evolution and the one-dimensional ball law

$a_n(x,y)=\exp\{\beta c_n(x,y)\}.$7

for the aggregate generated by $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$8 particles started at the origin. These identities establish the equality between several conservative and driven–dissipative definitions of the critical density in dimension one (Forien, 4 Feb 2025).

The point-source perspective sharpens this further. If $a_n(x,y)=\exp\{\beta c_n(x,y)\}.$9 is the stabilized configuration obtained from $\lambda$0 active particles at the origin, then for every sequence $\lambda$1 satisfying

$\lambda$2

one has

$\lambda$3

Moreover, any subsequential weak limit of the laws of $\lambda$4 on $\lambda$5 is shift invariant. Thus, throughout any macroscopic bulk window around the source, the local sleeping density is asymptotically the critical density, which is a precise local signature of self-organized criticality (Hoffman et al., 12 Jan 2026).

Finite-network ARW also admits a sharp relaxation theory. For the Markov chain on stable configurations obtained by adding one active particle with initial law $\lambda$6 and stabilizing under a killed random walk kernel $\lambda$7, let

$\lambda$8

Then the relaxation time is exactly

$\lambda$9

and separation cutoff occurs if and only if the product condition $P(x,a,t)$0 holds. The separation window satisfies

$P(x,a,t)$1

and explicit cutoff statements follow for large finite subsets of non-amenable graphs, large vertex-transitive graphs with a single sink, wheel-like graphs, and discrete Euclidean balls. The analysis proceeds through an exact reduction of separation distance to the tail of an IDLA filling time (Bristiel et al., 2022).

6. Analytical themes, misconceptions, and open directions

Several analytical templates recur across the subject. Single-particle activation models use Markovian embedding in the augmented state $P(x,a,t)$2, renewal decompositions over returns to the activation site, and Laplace/generating-function identities for the non-activated propagator. Continuous-space models translate local activation into local-time terms in the generator and Fokker–Planck equation. Reinforced walks deploy martingales for ratios of local times, the Rubin time-lines construction, and, on general graphs, replicator dynamics for occupation densities. Interacting-particle ARW uses the Diaconis–Fulton Abelian representation, odometers, IDLA couplings, and, in high-dimensional critical-density arguments, dormitory hierarchies, colored loops, and ping-pong rallies between clusters (Brémont et al., 2023, Meunier et al., 2016, Tarrès, 2011, Asselah et al., 2022, Ráth et al., 7 May 2026, Bristiel et al., 2022).

A common misconception is that local activation has a uniform dynamical consequence. The corpus shows the opposite. One localized activation site can yield superdiffusive or trapped behavior depending on $P(x,a,t)$3 or $P(x,a,t)$4; directed reinforcement on $P(x,a,t)$5 produces ballistic escape, while the same mechanism on cyclic or higher-dimensional graphs produces eventual trapping on a directed circuit; finite ARW always stabilizes on finite graphs, yet the stabilization time can be polynomial up to logarithmic corrections in one regime and exponential in system size in another (Brémont et al., 2023, Erhard et al., 2019, Basu et al., 2017).

The open-problem landscape is correspondingly broad. For single-particle LARWs, the literature explicitly points to multiple or extended activation regions, continuous-space higher-dimensional treatments, interacting walkers, evolving environments, non-monotone activation, and statistical inference of $P(x,a,t)$6 and $P(x,a,t)$7 from trajectories (Brémont et al., 2023). For the Ant RW, open problems include polynomial reinforcement in place of exponential reinforcement, the full dichotomy between circuit trapping and escape on general infinite graphs, and systems of multiple interacting ants sharing the same directed crossing field (Erhard et al., 2019). For ARW and VARW, current directions include shape theorems on growing domains, infinite-graph limits, supercritical regimes, universality of critical density, and finer stationary- or cutoff-type questions beyond what present methods control (Ráth et al., 7 May 2026, Bristiel et al., 2022). The cumulative picture is that LARWs are best understood not as one model but as a technical program: local memory and local activation rules, coupled to random motion, generate sharply geometry-dependent large-scale behavior.