Self-Interacting Random Walks

• Self-interacting random walks are non-Markovian processes where each step depends on past visits, local times, and adaptive measures.
• They encompass models such as excited, self-repellent, and once-reinforced walks, offering insights into recurrence, transience, and scaling limits.
• Analytical approaches use martingale techniques, branching processes, and stochastic approximation to derive exact propagators and variance reduction.

Self-interacting random walks (SIRWs) are non-Markovian random walks whose transition law depends on the trajectory’s own past through quantities such as site or edge local times, empirical occupation measures, cookie stacks, visited territory, or adaptive edge weights. In the models represented here, the memory may be encoded by the number of previous visits to neighboring sites, by edge-crossing counts, by a history-dependent empirical distribution on a graph, or by a changing conductance environment determined by the walk itself. This class includes excited random walks or cookie walks, self-repellent and self-attractive walks, once-reinforced models, and interacting vertex-reinforced systems, and it supports a theory of recurrence and transience, localization, scaling limits, persistence exponents, exact propagators, and stochastic-approximation limits (Davis et al., 2015, Doshi et al., 2023, Brémont et al., 2024, Park et al., 2024).

1. Model classes and state augmentation

A broad formulation starts from finitely many step laws $\mu_1,\dots,\mu_k$ on $\mathbb R^d$, together with an adapted rule $\ell(i)\in\{1,\dots,k\}$ choosing at time $i$ which measure governs the next increment: $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$ Here the history dependence lies entirely in the rule $\ell$, so that even when every $\mu_j$ has mean zero, the resulting walk can be transient or recurrent depending on how the past selects future increments (Peres et al., 2012).

A second standard representation is the cookie-environment formalism for excited random walks. A cookie environment is a family

$$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$

and the walk satisfies

$$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$

The next-step law is therefore indexed by the local visit count at the current site. In the nearest-neighbor survey literature, this is the defining mechanism of excited random walks or cookie walks on $\mathbb Z^d$ (Davis et al., 2015, Kosygina et al., 2012).

A third representation uses local times on edges. In one dimension, a nearest-neighbor walk may jump from $\mathbb R^d$0 to $\mathbb R^d$1 with probabilities

$\mathbb R^d$2

where $\mathbb R^d$3 is the number of crossings of the edge $\mathbb R^d$4 up to time $\mathbb R^d$5. In the broader site-based formulation of long-range memory, one writes $\mathbb R^d$6, so the walker effectively leaves a persistent local potential that feeds back into later motion (Brémont et al., 2024, Barbier--Chebbah et al., 2021).

On finite graphs, SIRWs can be written as nonlinear Markov chains driven by the empirical occupation measure

$\mathbb R^d$7

For self-repellent random walks (SRRWs) built from a reversible base kernel $\mathbb R^d$8 with stationary law $\mathbb R^d$9, the transition kernel becomes

$\ell(i)\in\{1,\dots,k\}$0

Frequent past visits suppress future transitions, so the interaction is explicitly self-repellent (Doshi et al., 2023).

A further general envelope is the random walk in changing environment (RWCE), where

$\ell(i)\in\{1,\dots,k\}$1

When the edge-weights $\ell(i)\in\{1,\dots,k\}$2 depend on the trajectory, the RWCE is adaptive and includes reinforced and self-interacting walks as special cases (Park et al., 2024).

2. Analytical frameworks

The analysis of SIRWs typically proceeds by enlarging the state space and isolating a tractable drift or occupation process. For cookie random walks with non-nearest-neighbor jumps, a basic tool is the martingale

$\ell(i)\in\{1,\dots,k\}$3

where $\ell(i)\in\{1,\dots,k\}$4 is the total drift already consumed from cookies up to time $\ell(i)\in\{1,\dots,k\}$5, with

$\ell(i)\in\{1,\dots,k\}$6

This produces optional-stopping estimates, a proof that $\ell(i)\in\{1,\dots,k\}$7, and a cookie-environment process

$\ell(i)\in\{1,\dots,k\}$8

that is weak Feller and admits a stationary measure $\ell(i)\in\{1,\dots,k\}$9, leading to a $i$0-$i$1 law for right transience (Davis et al., 2015).

For one-dimensional excited random walks, branching processes with migration and regeneration times are central. The survey literature encodes downcrossings before a new maximum in a branching-like process, obtains tail asymptotics for regeneration increments, and then derives strong laws, ballisticity thresholds, stable laws, and functional limit theorems in $i$2 and $i$3 topologies (Kosygina et al., 2012).

Generalized Ray–Knight theory plays an analogous role for edge-local-time SIRWs. In the asymptotically-free setting, the walk is decomposed as

$i$4

with a martingale part and an accumulated drift. Directed edge local times generate branching-like processes whose scaling limits are squared Bessel processes, and these control the extrema and the total drift. This machinery is the backbone of the functional convergence to Brownian motion perturbed at extrema (BMPE) in the asymptotically-free regime and of the negative result in the polynomially self-repelling regime (Liu et al., 2024, Kosygina et al., 2022).

Other SIRW subclasses are governed by different structural devices. The range of a one-parameter family of self-interacting walks on an interval is analyzed through a dyadic renormalization that yields a de Rham functional equation for the limit law of the scaled range (1207.1245). Comparison theorems for arrow systems convert local domination of stacks of left/right instructions into global domination of maxima, limsup behavior, and recurrence properties for one-dimensional self-interacting walks (Holmes et al., 2011). On arbitrary graphs, slowly changing RWCEs are studied by electrical network theory and super/submartingales built from harmonic voltages on finite truncations (Park et al., 2024). For once-reinforced random walk beyond exchangeability, a new explicit change-of-measure identity compares ORRW to simple random walk and yields a polymer representation suited to large deviations, transience, and local-time formulas (Collevecchio et al., 4 Sep 2025). On finite graphs and complete graphs, empirical occupation measures are treated as stochastic-approximation iterates tracking smooth ODEs with strict Lyapunov functions (Doshi et al., 2023, Pires et al., 2020).

3. Recurrence, transience, and confinement

The sharpest one-dimensional recurrence/transience criterion in the supplied corpus concerns excited random walks with non-nearest-neighbor jumps. In the one-cookie special case, the walk behaves like a simple symmetric random walk on revisits, while on the first visit to each site the next step is an independent copy of an integer-valued random variable $i$5 with $i$6 and $i$7. The theorem states that the walk is recurrent if $i$8 and transient to the right if $i$9. In the general i.i.d. cookie-environment model with finitely many nonnegative-drift cookies per site and a zero-mean bounded aperiodic background jump law $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$0, the same threshold is expressed through

$X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$1

(Davis et al., 2015).

In the nearest-neighbor excited setting, the one-dimensional phase diagram is classically parameterized by the expected total stored drift $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$2: recurrent if $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$3, transient to the right if $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$4, and transient to the left if $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$5. Ballisticity occurs only at the stricter threshold $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$6, with $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$7 for $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$8 and $X_0=0,\qquad X_{i+1}=X_i+\xi^{\ell(i)}_{i+1}.$9 for $\ell$0 (Kosygina et al., 2012).

A different general mechanism arises when the walk chooses among finitely many centered increment laws. If $\ell$1 are $\ell$2-dimensional mean-zero measures in $\ell$3, $\ell$4, with $\ell$5 moments, then every walk generated by them and any adapted rule is transient. More generally, if there exists a matrix $\ell$6 such that

$\ell$7

for every covariance matrix $\ell$8, then every walk generated by $\ell$9 and any adapted rule is transient. In dimension $\mu_j$0, the paper gives the complete picture recorded in the abstract: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures (Peres et al., 2012).

Several one-dimensional models exhibit finite-range confinement rather than mere recurrence. In “Stuck Walks,” the transition bias is driven by the local stream

$\mu_j$1

and there is a decreasing critical sequence

$\mu_j$2

If $\mu_j$3, then $\mu_j$4, while if $\mu_j$5, then almost surely $\mu_j$6. In particular, for $\mu_j$7 the walk is almost surely not trapped on any finite interval of this type (Erschler et al., 2010).

The two-parameter locally self-interacting family with

$\mu_j$8

has an additional sticky regime. When $\mu_j$9 and $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$0, thresholds

$$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$1

govern trapping: if $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$2, then with positive probability the walk eventually remains stuck on exactly $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$3 consecutive sites; if $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$4, it almost surely does not get stuck on fewer than $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$5 sites. The same paper also records asymmetric regimes where, with positive probability, $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$6, $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$7, or the motion is ballistic (Erschler et al., 2010).

For adaptive changing environments on arbitrary locally finite connected graphs, a strong stability theorem states that if the RWCE is proper, bounded from above, and the resistances satisfy

$$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$8

then the walk inherits recurrence or transience from the initial weighted graph $$\omega=\{\omega_{x,j}\}_{x\in\mathbb Z,\; j\ge 1}\in \Omega := \big(M_1(\mathbb Z)\big)^{\mathbb Z\times \mathbb N},$$9. The paper is explicit that this condition is too restrictive for classical reinforced walks such as ORRW or LRRW, but it applies on any graph, even with cycles (Park et al., 2024). Beyond this slow-change regime, once-reinforced random walk on general graphs admits large deviations for the range at Donsker–Varadhan scale and is transient on all non-amenable graphs for small reinforcement; a central point is that ORRW is not partially exchangeable, so the analysis requires a new change-of-measure formula rather than classical exchangeability tools (Collevecchio et al., 4 Sep 2025).

4. Scaling limits, range laws, and exact distributions

A major line of work asks whether diffusively rescaled one-dimensional SIRWs converge to BMPE. For asymptotically-free walks, the answer is positive. If the weight satisfies

$$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$0

then

$$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$1

in $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$2, where $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$3 is the pathwise unique solution of

$$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$4

The proof identifies the drift coefficient $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$5 through asymptotic imbalance in the weight function and shows that the accumulated drift is approximated by $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$6 times the signed range (Liu et al., 2024). A closely related result proves a full functional limit theorem for a large asymptotically-free class and emphasizes that this closes the gap left by earlier convergence only at geometric times (Kosygina et al., 2022).

The polynomially self-repelling regime behaves differently. For $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$7, $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$8, the diffusively rescaled walk does not converge to the BMPE predicted by the generalized Ray–Knight theorem and, more strongly, does not converge to any BMPE at all. This provides a counterexample to the idea that Ray–Knight convergence of local times is sufficient for functional convergence of the position process (Kosygina et al., 2022).

Range observables lead to a different type of scaling law. For a one-parameter family of self-interacting nearest-neighbor walks on $$P_\omega^x\!\left(X_{n+1}=X_n+z \,\middle|\, (X_k)_{k\le n}\right) =\omega_{X_n,L_n(X_n)}(z), \qquad L_n(y)=\sum_{k=0}^n \mathbf 1_{\{X_k=y\}}.$$9, the range up to exit from $\mathbb Z^d$0,

$\mathbb Z^d$1

satisfies weak convergence of $\mathbb Z^d$2 to a law on $\mathbb Z^d$3 whose distribution function $\mathbb Z^d$4 obeys a de Rham functional equation. The limit law is absolutely continuous when $\mathbb Z^d$5, singular when $\mathbb Z^d$6, and partly atomic for sufficiently strong interaction: if $\mathbb Z^d$7, every dyadic rational in $\mathbb Z^d$8 is an atom (1207.1245).

The propagator itself can sometimes be obtained exactly. For the once-reinforced walk denoted SATW in the exact-propagator paper, the scaling form is

$\mathbb Z^d$9

and the paper derives an explicit closed-form series for $\mathbb R^d$00. This yields the mean displacement $\mathbb R^d$01, a diffusion coefficient $\mathbb R^d$02 through

$\mathbb R^d$03

and a fourth cumulant of order $\mathbb R^d$04. The propagator is generally non-Gaussian, has Gaussian tails, and for sufficiently strong self-repulsion with $\mathbb R^d$05 can develop off-center maxima, which the paper interprets as an inherently non-Markovian mechanism pushing the walker away from its starting point (Brémont et al., 2024). The same work shows that the polynomially self-repelling walk has the exact propagator of the symmetric SATW with $\mathbb R^d$06 after the appropriate rescaling.

5. Persistence, aging, exploration, and first passage

Persistence theory for SIRWs is now sufficiently explicit to support universality-class statements. For nearest-neighbor walks whose transition probabilities depend on edge local times through a weight $\mathbb R^d$07, three classes are singled out: once-reinforced self-attracting walks $\mathbb R^d$08, polynomially self-repelling walks $\mathbb R^d$09, and sub-exponential self-repelling walks $\mathbb R^d$10. Their exact persistence exponents are

$\mathbb R^d$11

The derivation proceeds through exact splitting probabilities $\mathbb R^d$12, and for the TSAW case $\mathbb R^d$13 one obtains $\mathbb R^d$14 (Brémont et al., 2024).

Aging appears in a particularly explicit form in saturating SIRWs. If $\mathbb R^d$15 is the first hitting time of record level $\mathbb R^d$16 and $\mathbb R^d$17 is the record age, then the survival probability of the $\mathbb R^d$18-th record has the exact asymptotic scaling form

$\mathbb R^d$19

Two regimes coexist. For $\mathbb R^d$20,

$\mathbb R^d$21

where the paper emphasizes that record statistics are governed by the geometry of the explored region. For $\mathbb R^d$22,

$\mathbb R^d$23

so memory survives only through a prefactor and the tail exponent is fixed by the persistence exponent $\mathbb R^d$24. The hidden variable is the geometry of the visited interval, especially the random left boundary $\mathbb R^d$25 when the record $\mathbb R^d$26 is first reached (Brémont et al., 4 May 2026).

The broader exploration theory distinguishes compact and non-compact regimes. In infinite space, compact SIRWs have survival probability

$\mathbb R^d$27

whereas non-compact SIRWs satisfy $\mathbb R^d$28 and $\mathbb R^d$29, defining a transience exponent $\mathbb R^d$30. In confined domains of size $\mathbb R^d$31, first-passage times exhibit universal scaling forms in the large-domain limit. For compact fresh searches,

$\mathbb R^d$32

while for non-compact walks the mean first-passage time scales linearly with volume. The same framework argues that attractive self-interactions provide a decisive advantage for local space exploration, whereas repulsive self-interactions can significantly accelerate the global exploration of large domains (Barbier--Chebbah et al., 2021).

6. Nonlinear graph samplers and interacting systems

On finite graphs, self-repellent random walks furnish a fully analyzable SIRW model with direct algorithmic significance. For a reversible base kernel $\mathbb R^d$33 and target law $\mathbb R^d$34, the empirical measure obeys

$\mathbb R^d$35

under the compactness assumption stated in the paper. The associated ODE

$\mathbb R^d$36

has $\mathbb R^d$37 as unique fixed point, and the strict Lyapunov function

$\mathbb R^d$38

establishes global asymptotic stability. A central limit theorem gives an explicit asymptotic covariance matrix

$\mathbb R^d$39

and $\mathbb R^d$40 for all $\mathbb R^d$41, so larger repellence yields uniformly smaller asymptotic covariance. For SRRW-driven MCMC, the decrease in asymptotic sampling variance is of order $\mathbb R^d$42; the paper also notes an empirical tradeoff in which larger $\mathbb R^d$43 can slow early-time mixing, motivating time-varying schedules (Doshi et al., 2023).

The same nonlinear walk can drive distributed stochastic approximation. In SA-SRRW, the token update

$\mathbb R^d$44

is coupled to the empirical-measure update and to an optimization iterate

$\mathbb R^d$45

The paper proves $\mathbb R^d$46 almost surely and $\mathbb R^d$47 almost surely, together with a joint CLT in three timescale regimes. In the favorable regime $\mathbb R^d$48, the asymptotic covariance of the optimization error is always smaller than that of the base Markov-chain-driven algorithm and decays as $\mathbb R^d$49, amplifying the $\mathbb R^d$50 variance reduction known for the sampling-only SRRW. By contrast, case (iii), $\mathbb R^d$51, brings no asymptotic improvement (Hu et al., 2024).

Interacting reinforced systems supply a multi-particle analogue. On the complete graph with $\mathbb R^d$52 vertices, $\mathbb R^d$53 random walks choose vertices with probability exponentially biased by the occupation proportions of all walks. The empirical occupation measure satisfies a stochastic approximation recursion for the ODE $\mathbb R^d$54, and the strict Lyapunov function

$\mathbb R^d$55

implies almost sure convergence to the limit set of the flow and, under isolated equilibria, almost sure convergence to an equilibrium. If the interaction strengths are uniformly small, the equilibrium is unique; in the repelling two-walk, two-vertex example, the paper identifies a phase transition at $\mathbb R^d$56 from the symmetric equilibrium to two polarized stable equilibria (Pires et al., 2020).

Taken together, these graph-based models show that SIRWs are not confined to qualitative path properties such as recurrence, transience, or localization. They also define nonlinear samplers and adaptive stochastic-optimization mechanisms whose efficiency can be quantified exactly, while retaining the core feature that the next-step law depends on the process’s own empirical past (Doshi et al., 2023, Hu et al., 2024, Pires et al., 2020).