The true self-repelling motion: rigorous dynamics, the critical dimension d=2, and universality of the 3/2 local variation

The true self-repelling motion (TSRM) constructed by Tóth and Werner in 1998 is the canonical one-dimensional continuous process repelled by the density of its own occupation-time measure: it is self-similar with exponent $2/3$, Markov in the pair (position, local-time profile), and not a semimartingale — its paths have finite variation of order $3/2$. The 1998 paper left its program open on five fronts. The convergence side has since been settled: Newman and Ravishankar (2006) proved convergence of the discrete lattice-filling ('maze') model of the paper's Section 11; Horváth, Tóth and Vető (2012) settled the multidimensional question in the non-recurrent dimensions $d\ge3$; and Kosygina and Peterson (2026) proved the full functional invariance principle from the rescaled 'true' self-avoiding walk to the TSRM. The surviving open core is the analytic heart of the program: (1) a rigorous formulation, with uniqueness, of the locally self-repelling dynamics $dX_t = dB_t - \frac{\gamma}{2}\nabla L_t(X_t)\,dt$ — the one-dimensional self-repelling Brownian polymer was constructed rigorously only in 2025 (Giles–Gräfner) — together with the conjectural identification of its correctly rescaled law with the TSRM and the unproven one-dimensional asymptotics $\mathbb{E}|X_t|^2\sim t^{4/3}$ (currently between the Tauberian bounds $t^{5/4}$ and $t^{3/2}$); (2) the critical dimension $d=2$, where logarithmic superdiffusivity $\mathbb{E}|X_t|^2\sim t(\log t)^\beta$ with unknown $\beta$ is conjectured; and (3) the universality of the $3/2$ local-variation exponent for TSRM-type constructions from other coalescing reflected/absorbed diffusions. This core remains open: no resolution is known in the literature to date.

Background

Random walks that remember their own past form a canonical class of toy models for polymer growth and anomalous diffusion. The 'true' (or 'myopic') self-avoiding walk, introduced in the physics literature by Amit, Parisi and Peliti (1983), is a nearest-neighbour walk on $\mathbb{Z}^d$ that, at each step, prefers to step across the edges it has traversed least so far: the transition probability at time $n$ is a decreasing function of the local times (cumulative traversal counts) of the candidate edges. Unlike the standard self-avoiding walk, which is forbidden to revisit sites, the 'true' version only penalizes revisits, so it remains a Markov chain in the joint variable (position, local-time profile). In one dimension the displacement grows superdiffusively: Tóth (1995) proved, via a generalized Ray–Knight description of the walk's local-time process, that the position $S_n$ at time $n$ is of order $n^{2/3}$ (Brownian motion would give $n^{1/2}$) and identified the scaling limits of the marginals. (A Ray–Knight theorem is a statement describing the local-time process, viewed as a function of the space variable, at stopping times given by inverse local times; Tóth's generalized version does this for self-interacting walks, for which the local-time process itself is Markov.)

The continuous limit object was constructed by Tóth and Werner (1998) and named the true self-repelling motion (TSRM). Its construction is combinatorial rather than analytic: one takes a countable dense family of independent coalescing reflected/absorbed Brownian motions (RAB) in the half-plane $\{(x,h): h>0\}$ — an RAB started at $(x,h)$ is a Brownian motion with values in $[0,\infty)$, run with the horizontal coordinate as time, instantaneously reflected at the boundary $h=0$ up to time $0$ and killed at its first hitting of $0$ thereafter; the coalescing family is a close relative of the Brownian web, the continuum object describing all coalescing one-dimensional Brownian motions started from all space-time points, characterized by Fontes, Isopi, Newman and Ravishankar (2004). The RAB lines carve out a 'maze' in the plane, and the TSRM $X_t$ is the first coordinate of the unique curve $(X_t,H_t)$ that explores this maze while filling the plane, with $H_t$ tracking the height of the local-time profile at the current position. The process has an occupation-time density ('local time') $L_t(x)$ with trajectories continuous in $x$; the pair $(X_t, L_t(\cdot))$ is Markov; and the pair is self-similar with exponents $2/3$ and $1/3$: $(X_{at},H_{at})$ has the same law as $(a^{2/3}X_t, a^{1/3}H_t)$, so $X_t$ is of order $t^{2/3}$, matching the discrete exponent $n^{2/3}$. Strikingly, $X$ is not a semimartingale (roughly: not representable as a Brownian stochastic integral plus a finite-variation drift): its paths have finite variation of order $3/2$ — sums of $3/2$-powers of increments over fine partitions stay bounded while the quadratic variation diverges — which is impossible for nondegenerate semimartingales. Phenomenologically the dynamics is 'move away from where you have spent time': $dX_t$ tracks the negative spatial gradient of $L_t$ at the current position, in a generalized sense, plus boundary effects at the running extrema of the range; Tóth and Werner prove an averaged integral identity (their Theorem 6.1) but state explicitly that the pointwise differential equation 'does not make sense mathematically', because $x\mapsto L_t(x)$ is only as regular as a Brownian trajectory.

The 1998 paper explicitly leaves open: (i) the invariance principle — functional weak convergence of $n^{-2/3}S_{\lfloor nt\rfloor}$ to $X$ (Tóth and Werner indicated that finite-dimensional convergence should follow from the arguments of Tóth (1995) and deferred the delicate tightness arguments to future work; Kosygina and Peterson document that even the one-dimensional marginals, with the correct normalization, had not been rigorously established before their paper); (ii) a rigorous version of the differential formulation; (iii) multidimensional analogues ('much more difficult, and not treated here'); (iv) the convergence of the discrete Section 11 maze model; and (v) the suggestion that the $3/2$ local variation is 'in some sense universal' for constructions built from other coalescing diffusions — for example, reflected/absorbed Bessel processes of dimension $m\in(0,2)$ (the radial norm of an $m$-dimensional Brownian motion) raised to a power $a>0$, which have the self-similarity exponent $2/(2+a)$ in place of $2/3$ while being expected to share the $3/2$ local variation. The paper also records the authors' belief that TSRM is the scaling limit of other self-interacting processes, explicitly including the Brownian polymer of Durrett and Rogers (1992) — the self-repelling Brownian polymer (SRBP), formally $dX_t = dB_t - \frac{\gamma}{2}\nabla L_t(X_t)\,dt$ with $\gamma>0$, a continuous model of the same 'push away from your own occupation measure' dynamics, introduced in the probability literature by Durrett and Rogers and already written non-rigorously by Amit, Parisi and Peliti.

Two of these items are now closed by later work: Kosygina and Peterson (2026) proved the full functional invariance principle (the rescaled true self-avoiding walk and its local time converge, in the standard Skorokhod topology on the product space $D([0,\infty))^2$ of pairs of right-continuous paths with left limits, to $(X,H)$, by 'inverting' a joint generalized Ray–Knight theorem), and Newman and Ravishankar (2006) proved convergence of the rescaled Tóth lattice-filling curve (the Section 11 model) to the plane-filling curve. In dimensions $d\ge 3$ the story is also settled: Horváth, Tóth and Vető (2012) proved diffusive (Gaussian) central limit theorems for both the myopic self-avoiding walk and the SRBP, so no anomalous self-repelling motion survives there. What remains open is exactly the analytic heart of the program: the equation, the critical dimension $d=2$, and universality.

Problem Statement

The problem is to complete the rigorous analytic theory of the canonical locally self-repelling motion — the object constructed combinatorially in one dimension by Tóth and Werner (1998) — after the invariance principle and the lattice-filling convergence have been proven. Three linked components, each traceable to an open direction of the 1998 paper (the quantitative moment conjectures inside components (1) and (2) were made precise in the later literature, in the appendix of Tóth and Valkó, 2012):

(1) Equation and identification. The locally self-repelling dynamics is formally the singular stochastic differential equation $dX_t = dB_t - \frac{\gamma}{2}\nabla L_t(X_t)\,dt$ with coupling constant $\gamma>0$, where $B$ is a standard Brownian motion and $L_t$ is the occupation-time density of $X$ itself. In one dimension a solution of this equation was constructed rigorously, with a dynamic characterization of its law, only in 2025 (Giles and Gräfner). The open residue of this component is: (i) a rigorous analytic characterization with uniqueness of the Tóth–Werner motion itself, which is not a semimartingale (its paths have finite variation of order $3/2$) and is currently defined only through the combinatorial construction, its analytic handles being the averaged integral identity of Tóth–Werner's Theorem 6.1, the self-similarity exponents, and the explicit one-time marginal densities; and (ii) the identification conjecture stated by Tóth and Werner in 1998 and repeated in the current literature (Giles–Gräfner): under the correct superdiffusive rescaling, the one-dimensional self-repelling Brownian polymer converges to the Tóth–Werner true self-repelling motion. As a quantitative part of the same component, prove or refute the conjectured one-dimensional polymer asymptotics $\mathbb{E}|X_t|^2 \sim t^{4/3}$, which currently lies between the proven bounds $t^{5/4} \lesssim \mathbb{E}|X_t|^2 \lesssim t^{3/2}$, established in the Tauberian sense — that is, for Laplace transforms in time rather than directly for the moments (Tarrès–Tóth–Valkó).

(2) Critical dimension. Construct, or otherwise rigorously characterize, the locally self-repelling motion in $d=2$ — the only dimension where anomalous behaviour is expected but unproven ($d\ge3$ is diffusive by Horváth–Tóth–Vető, and the weak-coupling regime in $d=2$ is settled by Cannizzaro–Giles). Prove or refute the conjectured logarithmic superdiffusivity, at fixed coupling, of the two-dimensional self-repelling Brownian polymer (and, conjecturally, of its discrete counterpart, the two-dimensional true self-avoiding walk): $\mathbb{E}|X_t|^2 \sim t(\log t)^\beta$ for some $\beta\in(0,1)$, and determine the exponent $\beta$ if it exists (current evidence points to $\beta=1/2$).

(3) Universality. Prove or refute that the finite $3/2$ local variation (hence non-semimartingality) of the TSRM is universal for the class of Tóth–Werner-type constructions based on independent coalescing reflected/absorbed diffusions other than Brownian motion, in the sense made precise (but not proved) in Section 10 of Tóth and Werner (1998), which notes that such constructions have a different self-similarity exponent while being expected to share the $3/2$ local variation.

The three components concern one and the same object and are jointly needed to decide whether the true self-repelling motion is a well-defined universality class with a complete dimension-dependent anomalous-diffusion theory: superdiffusive $t^{2/3}$ scaling in $d=1$ (where the invariance principle is now proven), logarithmically superdiffusive $d=2$ (conjectural), and diffusive $d\ge3$ (proven). All three components are required for a complete solution; each sub-claim listed within a component may be settled independently (by proof or refutation), and settling any single one of them constitutes progress within that component but does not by itself resolve the task. Quantitative error estimates for the now-proven convergences (Kosygina–Peterson; Newman–Ravishankar) are a legitimate but secondary residue of the original list and are not part of this problem's core.

References

1. Bálint Tóth and Wendelin Werner, 'The true self-repelling motion', Probability Theory and Related Fields 111 (1998) 375–452. DOI: 10.1007/s004400050172. URL: https://doi.org/10.1007/s004400050172. (Primary source.)
2. Elena Kosygina and Jonathon Peterson, 'Convergence of rescaled "true" self-avoiding walks to the Tóth-Werner "true" self-repelling motion', Electronic Journal of Probability 31, Paper No. 32 (2026), 41 pp. DOI: 10.1214/26-EJP1489; arXiv:2502.10960. URL: https://arxiv.org/abs/2502.10960.
3. C. M. Newman and K. Ravishankar, 'Convergence of the Tóth lattice filling curve to the Tóth-Werner plane filling curve', ALEA. Latin American Journal of Probability and Mathematical Statistics 1 (2006) 333–346. No DOI assigned; zbMATH Open identifier 1110.65010 (document 5081211). URL: https://alea.impa.br/articles/v1/01-14.pdf. zbMATH record: https://zbmath.org/5081211.
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21. Elena Kosygina, Laure Marêché, Thomas Mountford and Jonathon Peterson, 'Extension of results on generalized Pólya's urns for polynomially self-repelling walks', preprint (2026), arXiv:2603.03622. URL: https://arxiv.org/abs/2603.03622. (A technical note announcing forthcoming scaling-limit work for the polynomially self-repelling family, which is a different model class.)