Self-Attracting Walk (SATW)

• Self-Attracting Walk (SATW) is a family of self-interacting stochastic processes where the future steps are biased by an attractive memory term based on already visited sites or edges.
• SATW models range from one-dimensional dynamic reinforcement and saturating walks to barycentric formulations and polymer measures, each displaying distinct asymptotic behaviors like ballistic motion, diffusive scaling, or localization.
• The framework spans both discrete lattice models and continuum diffusions, linking self-attraction with polymer physics, phase transitions, and critical phenomena while leaving key universality questions open.

Searching arXiv for relevant SATW papers and variants. Self-attracting walk (SATW) denotes a class of self-interacting stochastic processes in which the future motion is biased by the walk’s own past through an attractive memory term. In current usage, the label does not refer to a single canonical model. It appears for dynamically reinforced nearest-neighbor walks on $\mathbb Z$, centre-of-mass-interacting walks, self-avoiding walks with contact attraction, self-attracting polymer measures under an external drift, and self-attracting diffusions on compact manifolds (Agliari et al., 2012, Comets et al., 2010, Bauerschmidt et al., 2016, Ioffe et al., 2011, Gauthier, 2015). Across these formulations, attraction can be implemented through local times, nearest-neighbor contacts, adjacent plaquette rewards, or empirical averages, and the resulting asymptotic behavior includes ballistic motion, diffusive scaling with non-Gaussian propagators, logarithmic corrections at the upper critical dimension, and almost-sure convergence to a random limit.

1. Scope of the term and principal model classes

In the literature represented here, “self-attracting walk” is best understood as a family name for processes in which the path history enters the transition law or path weight with an attractive sign. The specific microscopic mechanism is model-dependent. In one dimension, attraction can be local and dynamic, with the transition probability at time $t$ depending on whether neighboring sites or edges have already been visited (Agliari et al., 2012, Brémont, 26 Jul 2025). In polymer formulations, attraction is encoded in an energy or Boltzmann weight that rewards contacts or adjacent parallel edges (Bauerschmidt et al., 2016, Hammond et al., 2017). In barycentric models, the walk is attracted to its own centre of mass (Comets et al., 2010). In continuum analogues, the drift is an integral over past positions (Gauthier, 2015).

Model class	Attractive mechanism	Representative result
True reinforced random walk with bias	Local-time reinforcement plus external field	Ballistic for every $s>0$ in $d=1$ (Agliari et al., 2012)
Saturating SATW	One-time bonus for crossing an unused edge	Always diffusive with exact non-Gaussian scaling laws (Brémont, 26 Jul 2025)
Barycentric SATW	Drift toward or away from $G_n=n^{-1}\sum_{i=1}^n X_i$	For $\beta\in[0,1)$, $n^{-1/(1+\beta)}X_n$ converges a.s. in the $\rho>0$ regime (Comets et al., 2010)
Weakly self-avoiding walk with contact attraction	Penalized self-intersections and rewarded nearest-neighbor contacts	In $d=4$, weak attraction preserves mean-field critical behavior with logarithmic corrections (Bauerschmidt et al., 2016)
Self-attracting SAW with bounded steps	Reward for adjacent parallel edges	In $d\ge5$, critical two-point function has mean-field decay (Hammond et al., 2017)
Self-attracting diffusion	Integral drift over the full past trajectory	On $t$0, $t$1 a.s. (Gauthier, 2015)

A recurring source of confusion is the assumption that all SATW models share a single universality class. The available results point in the opposite direction: the asymptotic regime depends sharply on whether attraction is local or global, saturating or cumulative, combined with self-avoidance, or opposed by an external drift.

2. One-dimensional reinforced and saturating SATWs

A basic lattice realization is the one-dimensional dynamic, or “true,” reinforced random walk with bias toward a target $t$2. If $t$3 is the position at time $t$4, and $t$5 or $t$6 according to whether site $t$7 has or has not been visited up to time $t$8, then with memory strength $t$9 and field strength $s>0$0,

$s>0$1

Before reaching the target, in the purely longitudinal regime,

$s>0$2

The dynamic path probability is locally normalized, in contrast with the static model in which the entire path carries a Boltzmann weight $s>0$3. For the dynamic model,

$s>0$4

In $s>0$5, the long-time behavior is explicit: for any $s>0$6,

$s>0$7

and

$s>0$8

Hence the walk is ballistic as soon as $s>0$9 is strictly positive, with no threshold $d=1$0; field effects prevail over memory effects without singularity. The paper contrasts this with the static model, where a true dynamical phase transition occurs at $d=1$1: for $d=1$2, $d=1$3, while for $d=1$4, $d=1$5 (Agliari et al., 2012).

The same work gives an approximate mapping, at $d=1$6, between the dynamic self-attracting walk and a trapping problem for a simple unbiased walk. After summing over the random border-visit variable $d=1$7,

$d=1$8

with

$d=1$9

This reformulates memory strength as an effective trap concentration.

A distinct one-dimensional SATW is the saturating model. On edges $G_n=n^{-1}\sum_{i=1}^n X_i$0, let

$G_n=n^{-1}\sum_{i=1}^n X_i$1

and define the reinforcement weight

$G_n=n^{-1}\sum_{i=1}^n X_i$2

If $G_n=n^{-1}\sum_{i=1}^n X_i$3, then

$G_n=n^{-1}\sum_{i=1}^n X_i$4

The attraction is therefore saturating: an unused edge receives an extra weight $G_n=n^{-1}\sum_{i=1}^n X_i$5, but once used, its weight becomes $G_n=n^{-1}\sum_{i=1}^n X_i$6 forever. In the scaling regime $G_n=n^{-1}\sum_{i=1}^n X_i$7, this model admits exact expressions for several observables. The splitting probability satisfies

$G_n=n^{-1}\sum_{i=1}^n X_i$8

where $G_n=n^{-1}\sum_{i=1}^n X_i$9, and the persistence exponent obeys $\beta\in[0,1)$0, so $\beta\in[0,1)$1. The propagator scales as

$\beta\in[0,1)$2

with an exact non-Gaussian density $\beta\in[0,1)$3; the span obeys

$\beta\in[0,1)$4

The thesis states that no dynamical phase transition occurs in the saturating SATW: the walk is always diffusive, with walk dimension $\beta\in[0,1)$5, while memory survives in non-Gaussian propagators, biased first-passage laws, persistence, and aged observables obtained through a Ray–Knight representation (Brémont, 26 Jul 2025).

3. Barycentric self-attracting walks

A different SATW class is defined through attraction to the walk’s own centre of mass. Let

$\beta\in[0,1)$6

Under Assumption (A1), jumps are bounded, uniformly elliptic, and have conditional mean drift

$\beta\in[0,1)$7

with $\beta\in[0,1)$8. The model is motivated as a random polymer chain in either poor or good solvent, and the summary presents $\beta\in[0,1)$9 as the attracting case (Comets et al., 2010).

The principal asymptotic classification is formulated for $n^{-1/(1+\beta)}X_n$0. If $n^{-1/(1+\beta)}X_n$1, then $n^{-1/(1+\beta)}X_n$2 is transient for $n^{-1/(1+\beta)}X_n$3 and recurrent for $n^{-1/(1+\beta)}X_n$4. If $n^{-1/(1+\beta)}X_n$5, with

$n^{-1/(1+\beta)}X_n$6

then $n^{-1/(1+\beta)}X_n$7 is recurrent for $n^{-1/(1+\beta)}X_n$8 and transient for $n^{-1/(1+\beta)}X_n$9. If $\rho>0$0, then $\rho>0$1 is recurrent in $\rho>0$2 and transient in $\rho>0$3.

In the regime $\rho>0$4, $\rho>0$5, the model has a super-diffusive law of large numbers. Writing

$\rho>0$6

there exists a random unit vector $\rho>0$7 such that almost surely

$\rho>0$8

$\rho>0$9

and

$d=4$0

For $d=4$1 this yields sub-ballistic escape. The summary also gives almost-sure upper bounds outside this regime: for $d=4$2,

$d=4$3

while for $d=4$4, $d=4$5,

$d=4$6

The analysis reduces $d=4$7 to a time-inhomogeneous Lamperti-type process with drift

$d=4$8

This places barycentric SATW at the intersection of self-interacting random walk, stochastic approximation, and non-Markovian recurrence theory (Comets et al., 2010).

4. Self-avoiding and polymer formulations

In polymer settings, self-attraction is often combined with self-avoidance rather than reinforcement. One bounded-step model on $d=4$9 assigns to each self-avoiding walk $d\ge5$0 the weight

$d\ge5$1

where $d\ge5$2 counts adjacent-edge pairs whose endpoints form a unit plaquette, $d\ge5$3 is induced by a symmetric spread-out step distribution $d\ge5$4, and $d\ge5$5 for non-self-avoiding walks. The partition function is

$d\ge5$6

the two-point function is

$d\ge5$7

and the susceptibility is $d\ge5$8. For each $d\ge5$9, there exists $t$00 such that for $t$01, the connective constant

$t$02

exists. In $t$03, for large spread-out parameter $t$04 and small $t$05, the critical two-point function satisfies

$t$06

so the critical exponent $t$07 (Hammond et al., 2017).

A continuous-time counterpart in $t$08 is the weakly self-avoiding walk with contact self-attraction. If $t$09, then

$t$10

and the energy is

$t$11

Self-intersections are penalized, while nearest-neighbor contacts are rewarded when $t$12. The two-point function,

$t$13

the susceptibility $t$14, and the correlation lengths $t$15 are controlled rigorously by a supersymmetric representation and a multi-scale renormalisation group. For sufficiently small $t$16 and $t$17,

$t$18

$t$19

and for each $t$20,

$t$21

These results show that small contact self-attraction leaves the $t$22 critical behavior in the same mean-field universality class as the pure weakly self-avoiding walk, with the Gaussian fixed point stable against small attractive perturbations. The summary states that a collapse transition is predicted for larger self-attraction, but its location and critical behavior remain open (Bauerschmidt et al., 2016).

A broader polymer measure with self-attraction and endpoint drift is defined by

$t$23

where $t$24 is non-decreasing and subadditive. There exists a compact convex set $t$25 such that the walk is sub-ballistic for $t$26 and ballistic for $t$27. The boundary $t$28 is the set of critical drifts. The principal result is that for every $t$29, the limiting velocity exists and is strictly positive; the sub-ballistic to ballistic transition is therefore first order. At criticality one also has a law of large numbers and a central limit theorem, with

$t$30

where $t$31 are i.i.d. irreducible pieces arising from a renewal decomposition (Ioffe et al., 2011).

5. Growing self-avoiding walks, trapping, and the $t$32-point controversy

A further self-attracting mechanism appears in growing self-avoiding walks on the square lattice. Starting at the origin, the walker chooses among empty nearest-neighbor sites only. If a prospective step $t$33 creates $t$34 new non-adjacent nearest-neighbor contacts, then

$t$35

For a full walk with total contact number $t$36, the Boltzmann weight is

$t$37

This is a kinetic growth ensemble rather than an equilibrium polymer measure (Hooper et al., 2020).

The central observable is the trapping length $t$38, the step at which no empty neighbor remains. Its mean is

$t$39

At zero attraction,

$t$40

For sufficiently strong attraction, $t$41,

$t$42

Numerically there is also a pronounced local minimum,

$t$43

The paper attributes this to a two-step trapping mechanism: weak attraction preferentially favors entry into an already formed void but does not yet suppress void formation itself. For the shortest trapped walk, $t$44, an exact expression is given for $t$45, and its maximum occurs near $t$46, paralleling the minimum of $t$47 (Hooper et al., 2020).

The same work addresses whether growing self-avoiding walks share the same asymptotic behavior as equilibrium self-avoiding walks. For the equilibrium two-dimensional SAW with nearest-neighbor attraction, the $t$48-point occurs at

$t$49

with universal amplitude ratio

$t$50

In the growing ensemble, however, no stable crossing near $t$51 is observed; the data instead suggest

$t$52

and drifting upward as the maximal walk length increases because of attrition by trapping. The summary therefore states

$t$53

The same section reports the expected values

$t$54

but only at attractions well above the equilibrium $t$55-point. The persistence length,

$t$56

converges rapidly by $t$57, with

$t$58

and decreases smoothly toward the random-walk limit $t$59 as $t$60 increases (Hooper et al., 2020).

6. Self-attracting diffusions on the sphere and circle

Self-attraction also has a continuum formulation. On the unit sphere $t$61, with $t$62 and $t$63, the self-attracting diffusion is the Stratonovich SDE

$t$64

Since

$t$65

the drift term is an attraction toward past positions in the tangent direction. The SDE has a unique strong solution. The principal theorem states that there exists an $t$66-valued random variable $t$67 such that almost surely

$t$68

In particular, $t$69 almost surely (Gauthier, 2015).

When $t$70, writing $t$71, the equation reduces to

$t$72

The corresponding theorem gives almost-sure convergence to a random angle $t$73, with

$t$74

The proof introduces the averaged past position

$t$75

derives

$t$76

and a closed SDE for $t$77, then shows $t$78, $t$79, and $t$80, from which $t$81 follows. This places self-attracting diffusion within the theory of asymptotic pseudotrajectories and stochastic dynamics on manifolds (Gauthier, 2015).

7. Related localized walks and open directions

Self-attraction on $t$82 is closely related to other locally self-interacting walks in which attraction competes with repulsion. In the “stuck walks” model, with edge local times $t$83, the local stream is

$t$84

and the jumps satisfy

$t$85

The critical sequence

$t$86

governs localization windows. If $t$87, then the original theorem gives positive probability of confinement to an interval of $t$88 consecutive sites; if $t$89, trapping on $t$90 sites does not occur (Erschler et al., 2010). A later result proves that under the same assumption,

$t$91

and in the special case $t$92, the walk localizes almost surely on $t$93 sites (Kious, 2013). These models are not usually labeled SATW, but they show that local attractive memory can induce genuine finite-range localization.

Several open directions remain explicit in the cited works. For weakly self-avoiding walk with contact attraction in $t$94, the location and critical behavior of the predicted collapse point $t$95 are left open (Bauerschmidt et al., 2016). For bounded-range self-attracting self-avoiding walk, the finite-range $t$96 case remains open (Hammond et al., 2017). For stuck walks, full almost-sure localization on exactly $t$97 sites is conjectured, while exclusion of the $t$98-site case is unfinished in general (Kious, 2013). For the growing self-avoiding walk with attraction, strong finite-size effects obstruct a clean determination of the $t$99-point (Hooper et al., 2020). The 2025 thesis states that it includes the first exact expressions for aged observables in the saturating SATW model, which suggests that two-time non-equilibrium statistics are becoming a central diagnostic alongside one-time observables such as propagators and first-passage laws (Brémont, 26 Jul 2025).

Taken together, these results suggest that “self-attraction” is a mechanism rather than a single phase or universality class. Depending on whether the attraction is dynamic or static, saturating or cumulative, local or barycentric, lattice-based or continuous, it can produce ballistic transport, diffusive exploration with exact non-Gaussian laws, localization, or polymeric mean-field criticality with logarithmic corrections.