--- title: Self-Attracting Walk (SATW) url: https://www.emergentmind.com/topics/self-attracting-walk-satw type: topic --- # Self-Attracting Walk (SATW) 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 [1206.1550] [1003.3121] [1610.08573] [1104.4615] [1501.04827]. 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 [1206.1550] [2507.19937]. In polymer formulations, attraction is encoded in an energy or Boltzmann weight that rewards contacts or adjacent parallel edges [1610.08573] [1712.07673]. In barycentric models, the walk is attracted to its own centre of mass [1003.3121]. In continuum analogues, the drift is an integral over past positions [1501.04827]. | Model class | Attractive mechanism | Representative result | |---|---|---| | True reinforced random walk with bias | Local-time reinforcement plus external field | Ballistic for every \(s>0\) in \(d=1\) [1206.1550] | | Saturating SATW | One-time bonus for crossing an unused edge | Always diffusive with exact non-Gaussian scaling laws [2507.19937] | | 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 [1003.3121] | | 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 [1610.08573] | | Self-attracting SAW with bounded steps | Reward for adjacent parallel edges | In \(d\ge5\), critical two-point function has mean-field decay [1712.07673] | | Self-attracting diffusion | Integral drift over the full past trajectory | On \(S^n\), \(X_t\to X_\infty\) a.s. [1501.04827] | 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 \(T\). If \(x_t\in\mathbb Z\) is the position at time \(t\), and \(k_t(j)=1\) or \(0\) according to whether site \(j\) has or has not been visited up to time \(t\), then with memory strength \(u\) and field strength \(s\), \[ p(x_t\to x_t\pm1)=\frac{\exp[u\,k_t(x_t\pm1)-s|T-(x_t\pm1)|]}{\exp[u\,k_t(x_t+1)-s|T-(x_t+1)|]+\exp[u\,k_t(x_t-1)-s|T-(x_t-1)|]}. \] Before reaching the target, in the purely longitudinal regime, \[ p(x_t\to x_t\pm1)=\frac{\exp[u\,k_t(x_t\pm1)\pm s]}{\exp(u\,k_t(x_t+1)+s)+\exp(u\,k_t(x_t-1)-s)}. \] The dynamic path probability is locally normalized, in contrast with the static model in which the entire path carries a Boltzmann weight \(P_{\mathrm{stat}}(w)\propto \exp[-u\,S(w)+s\,x(w)]\). For the dynamic model, \[ P_{\mathrm{dyn}}(w)=\frac{\exp[-u\,S_t+s\,x_t]}{(2\cosh s)^t\,C(t_R,t_L)}. \] In \(d=1\), the long-time behavior is explicit: for any \(s>0\), \[ \langle S_t\rangle\sim v(s,u)\,t,\qquad v(s,u)=\frac{e^{2s}-1}{e^{2s}-1+2e^u} =\frac{1}{1+e^u(1/\tanh s-1)}, \] and \[ \langle x_t\rangle\sim t\,\tanh s. \] Hence the walk is ballistic as soon as \(s\) is strictly positive, with no threshold \(s_c>0\); field effects prevail over memory effects without singularity. The paper contrasts this with the static model, where a true dynamical phase transition occurs at \(s=u\): for \(su\), \(v=\tanh(s-u)>0\) [1206.1550]. The same work gives an approximate mapping, at \(s=0\), between the dynamic self-attracting walk and a trapping problem for a simple unbiased walk. After summing over the random border-visit variable \(t_B\), \[ P_{u,0}(S,t)\simeq P_{0,0}(S,t)\,e^{-u'S}, \] with \[ u'=u-\frac{2(1-e^{-u})}{1+e^{-u}},\qquad c=1-e^{-u'}. \] This reformulates memory strength as an effective trap concentration. A distinct one-dimensional SATW is the saturating model. On edges \(\{x-1,x\}\), let \[ L_t(x)=\#\{s0,\qquad w(n\ge1)=1. \] If \(X_t=x\), then \[ P(X_{t+1}=x\pm1\mid X_t=x)=\frac{w(L_t(x\pm1))}{w(L_t(x+1))+w(L_t(x-1))}. \] The attraction is therefore saturating: an unused edge receives an extra weight \(\phi\), but once used, its weight becomes \(1\) forever. In the scaling regime \(x^2\sim t\), this model admits exact expressions for several observables. The splitting probability satisfies \[ Q_+(z)=I_{1-z}(\phi), \] where \(z=k/(k+m)\), and the persistence exponent obeys \(\phi=2\theta\), so \(\theta=\phi/2\). The propagator scales as \[ P(x,t)\sim (2t)^{-1/2}p_\phi(u),\qquad u=x/\sqrt{2t}, \] with an exact non-Gaussian density \(p_\phi\); the span obeys \[ \langle N(t)\rangle\sim \frac{\Gamma(\phi)}{\Gamma(\phi+1/2)}\sqrt{2t}, \qquad \mathrm{Var}(N(t))=C(\phi)\,t. \] The thesis states that no dynamical phase transition occurs in the saturating SATW: the walk is always diffusive, with walk dimension \(d_w=2\), while memory survives in non-Gaussian propagators, biased first-passage laws, persistence, and aged observables obtained through a Ray–Knight representation [2507.19937]. ## 3. Barycentric self-attracting walks A different SATW class is defined through attraction to the walk’s own centre of mass. Let \[ G_n=\frac1n\sum_{i=1}^n X_i,\qquad Y_n=X_n-G_n,\qquad \Delta_n=X_{n+1}-X_n. \] Under Assumption (A1), jumps are bounded, uniformly elliptic, and have conditional mean drift \[ E[\Delta_n\mid\mathcal F_n] = \rho\,\|Y_n\|^{-\beta}\,\hat Y_n +O\!\left(\|Y_n\|^{-\beta}(\log\|Y_n\|)^{-2}\right), \] with \(\hat Y_n=Y_n/\|Y_n\|\). The model is motivated as a random polymer chain in either poor or good solvent, and the summary presents \(\rho>0\) as the attracting case [1003.3121]. The principal asymptotic classification is formulated for \(Y_n=X_n-G_n\). If \(\beta\in[0,1)\), then \(Y_n\) is transient for \(\rho>0\) and recurrent for \(\rho<0\). If \(\beta=1\), with \[ \rho_0(d,\sigma^2)=\frac{(2-d)\sigma^2}{2}, \] then \(Y_n\) is recurrent for \(\rho\le \rho_0\) and transient for \(\rho>\rho_0\). If \(\beta>1\), then \(Y_n\) is recurrent in \(d\le2\) and transient in \(d\ge3\). In the regime \(\beta\in[0,1)\), \(\rho>0\), the model has a super-diffusive law of large numbers. Writing \[ \ell(\rho,\beta)=\left[\frac{\rho(1+\beta)}{2+\beta}\right]^{1/(1+\beta)}, \] there exists a random unit vector \(u\in S^{d-1}\) such that almost surely \[ n^{-1/(1+\beta)}X_n\to (2+\beta)^{1/(1+\beta)}\ell(\rho,\beta)\,u, \] \[ n^{-1/(1+\beta)}G_n\to (1+\beta)\ell(\rho,\beta)\,u, \] and \[ n^{-1/(1+\beta)}Y_n\to \ell(\rho,\beta)\,u. \] For \(\beta\in(0,1)\) this yields sub-ballistic escape. The summary also gives almost-sure upper bounds outside this regime: for \(\beta\ge1\), \[ \|X_n\|\le C\,n^{1/2}(\log n)^{1/2+\varepsilon} \quad\text{a.s. for large }n, \] while for \(\beta\in[0,1)\), \(\rho<0\), \[ \|X_n\|\le (\log n)^{1+1/(1-\beta)+\varepsilon} \quad\text{a.s. for large }n. \] The analysis reduces \(\|Y_n\|\) to a time-inhomogeneous Lamperti-type process with drift \[ \mu_1(n;Z_n)=\rho Z_n^{-\beta}-\frac{Z_n}{n}+o(Z_n^{-\beta}+Z_n/n), \qquad Z_n=\|Y_n\|. \] This places barycentric SATW at the intersection of self-interacting random walk, stochastic approximation, and non-Markovian recurrence theory [1003.3121]. ## 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 \(\mathbb Z^d\) assigns to each self-avoiding walk \(w\) the weight \[ W_K(w)=P_n(w)\,(1+K)^{|\mathrm{adj}(w)|}, \] where \(\mathrm{adj}(w)\) counts adjacent-edge pairs whose endpoints form a unit plaquette, \(P_n(w)\) is induced by a symmetric spread-out step distribution \(D\), and \(W_K(w)=0\) for non-self-avoiding walks. The partition function is \[ C_n(K)=\sum_{w\in\mathcal I_n}W_K(w), \] the two-point function is \[ G_{z,K}(x)=\sum_{n\ge0}z^n\sum_{w\in\mathcal I_n(0,x)}W_K(w), \] and the susceptibility is \(X_K(z)=\sum_x G_{z,K}(x)\). For each \(d\ge2\), there exists \(K_0>0\) such that for \(00\). The two-point function, \[ G_{\beta,\gamma,\nu}(a,b)=\int_0^\infty c_T(a,b)e^{-\nu T}\,dT, \] the susceptibility \(\chi(\beta,\gamma,\nu)\), and the correlation lengths \(\xi_p(\beta,\gamma,\nu)\) are controlled rigorously by a supersymmetric representation and a multi-scale renormalisation group. For sufficiently small \(0<\beta<\beta_*\) and \(|\gamma|<\gamma_*(\beta)\), \[ G_{\beta,\gamma,\nu_c}(0,x)=A_{\beta,\gamma}|x|^{-2}\Bigl(1+O(1/\log|x|)\Bigr), \] \[ \chi(\beta,\gamma,\nu_c+\varepsilon)\sim B_{\beta,\gamma}\,\varepsilon^{-1}\,(\log(1/\varepsilon))^{1/4}, \] and for each \(p>0\), \[ \xi_p(\beta,\gamma,\nu_c+\varepsilon)\sim B_{\beta,\gamma}^{1/2}c_p\,\varepsilon^{-1/2}\,(\log(1/\varepsilon))^{1/8}. \] These results show that small contact self-attraction leaves the \(d=4\) 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 [1610.08573]. A broader polymer measure with self-attraction and endpoint drift is defined by \[ a_{h,\lambda}(\gamma)=\exp\{-\Phi(\gamma)+h\cdot D(\gamma)-\lambda|\gamma|\}, \qquad \Phi(\gamma)=\sum_x \phi(\ell_\gamma[x]), \] where \(\phi\) is non-decreasing and subadditive. There exists a compact convex set \(K\subset\mathbb R^d\) such that the walk is sub-ballistic for \(h\in\mathrm{int}\,K\) and ballistic for \(h\notin K\). The boundary \(\partial K\) is the set of critical drifts. The principal result is that for every \(h\in\partial K\), 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 \[ v(h)=\frac{E_h[X]}{E_h[T]}, \qquad \Sigma(h)=\frac{1}{E_h[T]}\mathrm{Var}_h[X-v(h)T], \] where \((T_i,X_i)\) are i.i.d. irreducible pieces arising from a renewal decomposition [1104.4615]. ## 5. Growing self-avoiding walks, trapping, and the \(\theta\)-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 \(i\) creates \(m_i\) new non-adjacent nearest-neighbor contacts, then \[ F_i=-\epsilon\,m_i,\qquad Z=\sum_{i=1}^n e^{\beta\epsilon m_i},\qquad p_i=\frac{e^{\beta\epsilon m_i}}{Z}. \] For a full walk with total contact number \(k\), the Boltzmann weight is \[ W(\epsilon,k)=\exp(\beta\epsilon k). \] This is a kinetic growth ensemble rather than an equilibrium polymer measure [2006.12680]. The central observable is the trapping length \(N\), the step at which no empty neighbor remains. Its mean is \[ \langle N(\epsilon)\rangle=\sum_{N=7}^\infty N\,P_{\mathrm{trap}}(N;\epsilon). \] At zero attraction, \[ \langle N(0)\rangle=70.85\pm0.05. \] For sufficiently strong attraction, \(\beta\epsilon\gtrsim1\), \[ \langle N(\epsilon)\rangle \simeq A\,\exp(B\,\beta\epsilon), \qquad B\approx1.12. \] Numerically there is also a pronounced local minimum, \[ \min_\epsilon \langle N(\epsilon)\rangle \approx 67.2 \quad\text{at}\quad \beta\epsilon\approx0.28. \] 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, \(N=7\), an exact expression is given for \(P_7(\epsilon)\), and its maximum occurs near \(\beta\epsilon\approx0.234\), paralleling the minimum of \(\langle N\rangle\) [2006.12680]. 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 \(\theta\)-point occurs at \[ \beta\epsilon_\theta^{(\mathrm{SAW})}\approx0.66, \qquad \nu_\theta=4/7\approx0.571, \] with universal amplitude ratio \[ \langle R_g^2\rangle/\langle R_{ee}^2\rangle\approx0.18. \] In the growing ensemble, however, no stable crossing near \(\beta\epsilon=0.66\) is observed; the data instead suggest \[ \beta\epsilon_\theta^{(\mathrm{GSAW})}>3, \] and drifting upward as the maximal walk length increases because of attrition by trapping. The summary therefore states \[ \beta\epsilon_\theta^{(\mathrm{GSAW})}\ne \beta\epsilon_\theta^{(\mathrm{SAW})}. \] The same section reports the expected values \[ \nu_{\mathrm{swollen}}\approx0.68<0.75,\qquad \nu_{\mathrm{collapsed}}=0.5,\qquad \nu_\theta\approx 4/7, \] but only at attractions well above the equilibrium \(\theta\)-point. The persistence length, \[ \ell_p(N;\epsilon)=\langle \mathbf r_N\cdot \hat{\mathbf e}_1\rangle, \] converges rapidly by \(N\sim30\), with \[ \ell_p(0)\approx1.402\pm0.001, \] and decreases smoothly toward the random-walk limit \(0.5\) as \(\beta\epsilon\) increases [2006.12680]. ## 6. Self-attracting diffusions on the sphere and circle Self-attraction also has a continuum formulation. On the unit sphere \(S^n\subset\mathbb R^{n+1}\), with \(\sigma>0\) and \(a<0\), the self-attracting diffusion is the Stratonovich SDE \[ dX_t = \sigma\,dW_t(X_t) - a\int_0^t \nabla_{S^n}V_{X_s}(X_t)\,ds\,dt, \qquad V_y(x)=\langle x,y\rangle. \] Since \[ \nabla_{S^n}V_y(x)=y-\langle x,y\rangle x, \] 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 \(S^n\)-valued random variable \(X_\infty\) such that almost surely \[ \|X_t-X_\infty\| = O\!\left(t^{-1/2}\sqrt{\log^\gamma t}\right) \quad\text{for any }\gamma>1. \] In particular, \(X_t\to X_\infty\) almost surely [1501.04827]. When \(n=1\), writing \(X_t=(\cos\theta_t,\sin\theta_t)\), the equation reduces to \[ d\theta_t = \sigma\,dW_t + a\int_0^t \sin(\theta_t-\theta_s)\,ds\,dt. \] The corresponding theorem gives almost-sure convergence to a random angle \(\theta_\infty\in\mathbb R/2\pi\mathbb Z\), with \[ |\theta_t-\theta_\infty| = O\!\left(t^{-1/2}\sqrt{\log^\gamma t}\right) \quad\text{a.s.} \] The proof introduces the averaged past position \[ U_t=\int_0^t X_s\,ds,\qquad R_t=\|U_t\|,\qquad V_t=U_t/R_t,\qquad O_t=\langle V_t,X_t\rangle, \] derives \[ dR_t=O_t\,dt, \] and a closed SDE for \(O_t\), then shows \(R_t\to\infty\), \(O_t\to1\), and \(V_t\to V_\infty\), from which \(X_t\to V_\infty\) follows. This places self-attracting diffusion within the theory of asymptotic pseudotrajectories and stochastic dynamics on manifolds [1501.04827]. ## 7. Related localized walks and open directions Self-attraction on \(\mathbb Z\) is closely related to other locally self-interacting walks in which attraction competes with repulsion. In the “stuck walks” model, with edge local times \(L(n,j)\), the local stream is \[ \Delta(n,x)= -\alpha\,L(n,x-1)+L(n,x)-L(n,x+1)+\alpha\,L(n,x+2), \] and the jumps satisfy \[ P(X_{n+1}=X_n\pm1\mid\mathcal F_n) = \frac{\exp\{\pm\beta\Delta_n\}}{\exp\{\beta\Delta_n\}+\exp\{-\beta\Delta_n\}}. \] The critical sequence \[ \alpha_1=+\infty,\qquad \alpha_L=\frac{1}{1+2\cos(2\pi/(L+2))},\quad L\ge2, \] governs localization windows. If \(\alpha\in(\alpha_{L+1},\alpha_L)\), then the original theorem gives positive probability of confinement to an interval of \(L+2\) consecutive sites; if \(\alpha<\alpha_{L+1}\), trapping on \(L+2\) sites does not occur [1011.1103]. A later result proves that under the same assumption, \[ |R'|\in\{L+2,L+3\}\quad\text{a.s.}, \] and in the special case \(\alpha>1\), the walk localizes almost surely on \(3\) sites [1309.1586]. 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 \(d=4\), the location and critical behavior of the predicted collapse point \(\gamma_\theta(\beta)\) are left open [1610.08573]. For bounded-range self-attracting self-avoiding walk, the finite-range \(d=4\) case remains open [1712.07673]. For stuck walks, full almost-sure localization on exactly \(L+2\) sites is conjectured, while exclusion of the \(L+3\)-site case is unfinished in general [1309.1586]. For the growing self-avoiding walk with attraction, strong finite-size effects obstruct a clean determination of the \(\theta\)-point [2006.12680]. 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 [2507.19937]. 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. Source: https://www.emergentmind.com/topics/self-attracting-walk-satw