Self-Repelling Brownian Polymer

• The self-repelling Brownian polymer is a stochastic process where the path is repelled from regions it has previously visited, embodying key characteristics of excluded-volume effects.
• It displays dimension-dependent behavior, with diffusive motion in higher dimensions, logarithmically superdiffusive scaling in two dimensions, and superdiffusivity in one dimension.
• Analytical techniques such as martingale approximations, Fock space representations, and resolvent calculus are employed to elucidate its scaling laws, ergodicity, and convergence properties.

A self-repelling Brownian polymer is a continuous-time, continuous-space stochastic process in which the trajectory is repelled from regions it has previously visited. The self-repulsion typically manifests through a history-dependent drift proportional to the (negative) gradient of the local time or an occupation measure of the process. Such processes serve as canonical models for polymers with excluded-volume effects and self-avoidance, and display phenomena ranging from diffusive to superdiffusive behavior depending on the spatial dimension and specifics of the self-repulsive interaction.

1. Mathematical Formulation of Self-Repelling Brownian Polymers

The prototypical self-repelling Brownian polymer (SRBP) in $\mathbb{R}^d$ is given as the solution to the stochastic differential equation

$dX_t = dB_t - \nabla(V * \ell_t)(X_t)\,dt$

where $B_t$ is standard Brownian motion, $\ell_t$ is the local time or occupation measure up to time $t$, and $V$ is a smooth, positive-definite, fast-decaying kernel (e.g., a mollified delta function) (Horvath et al., 2010).

The occupation measure is defined as

$\ell_t(A) = \int_0^t \mathbb{1}_{X_s \in A}\,ds$

for measurable $A \subseteq \mathbb{R}^d$. The convolution $V * \ell_t$ smooths the (singular) local time, making the resulting drift a continuous function for all $t > 0$ provided $V$ is sufficiently regular.

In 1D, the singular model

$$dX_t = dB_t - \beta^2 \left(\frac{d}{dx} L_t\right)(X_t)\,dt$$

where $L_t(x)$ is local time at $x$, requires renormalization and technical constructions using the theory of singular stochastic partial differential equations and energy solutions (Giles et al., 5 Sep 2025).

In broader contexts, models such as self-repelling fractional Brownian motion, self-repelling diffusions on Riemannian manifolds, or active Brownian polymers with self-avoidance (excluded-volume effects) extend this framework (Bornales et al., 2011, Benaïm et al., 2015, Eleutério et al., 21 Aug 2024).

2. Scaling Laws: Diffusive, Superdiffusive, and Logarithmic Corrections

The scaling behavior of the mean-square displacement (MSD) of self-repelling Brownian polymers exhibits dimension-specific regimes:

• $d \geq 3$: The process is diffusive, i.e.,

$\mathbb{E}|X_t|^2 \sim \sigma^2 t$

for some $\sigma^2 > 0$ (Horvath et al., 2010). A full central limit theorem (CLT) holds for the rescaled trajectory: $(1/\sqrt{N}) X(N t)$ converges in finite-dimensional distributions to a Brownian motion with variance $\sigma^2$.

• $d=2$: The process is (logarithmically) superdiffusive, with rigorous bounds

$$c_1\, t\log\log t \leq \mathbb{E}|X_t|^2 \leq c_2\, t\log t$$

and non-rigorous scaling arguments predicting

$\mathbb{E}|X_t|^2 \sim t (\log t)^{1/2}$

(Toth et al., 2010, Cannizzaro et al., 11 Mar 2024). Recent results demonstrate that upon scaling both time and the coupling constant ("weak coupling scaling"), the limiting process is Brownian with an enhanced diffusivity; the correction to diffusivity provides compelling evidence for the $(\log t)^{1/2}$ scaling conjecture (Cannizzaro et al., 11 Mar 2024).

• $d=1$: The process is strictly superdiffusive. In the rigorous sense,

$$t^{5/4} \lesssim \mathbb{E}[X_t^2] \lesssim t^{3/2}$$

(in Tauberian sense), confirming that the history-dependent memory effect leads to faster-than-diffusive spread (Giles et al., 5 Sep 2025).

• Polynomially decaying self-repulsion (varying the weight of the drift as a function of past occupation): For so-called "asymptotically free" self-interacting random walks, functional limit theorems show convergence to Brownian motion perturbed at extrema (BMPE). For stronger, polynomially decaying self-repulsion, the process does not even converge in any diffusive scaling (Kosygina et al., 2022).

3. Martingale-Compensator Structure and Analytical Techniques

The analysis leverages a decomposition of the displacement,

$X_t = B_t + \int_0^t F(X_s, \ell_s)\,ds$

with the second term ("compensator") reflecting the accumulated self-repulsive force. In the diffusive regime ($d \geq 3$), one applies non-reversible Kipnis–Varadhan theory with a graded sector condition to establish CLTs (Horvath et al., 2010). The method involves:

• Resolvent calculus for non-reversible generators,
• Martingale approximations (the compensator term approximates an additive functional of Markov process in the environment seen from the particle),
• Variational bounds on $H_{-1}$-norms of additive functionals,
• Fock space representations and explicit operator computations in the Gaussian Hilbert space of the environment.

In two dimensions, superdiffusive behavior is proven using resolvent methods and variational inequalities for the Laplace transform of the MSD; non-rigorous scaling arguments using Green–Kubo formulas support the logarithmic correction exponents (Toth et al., 2010).

In one dimension, the construction via mollified approximations and energy solutions to singular SPDEs makes the model well-posed despite the singular distributional drift (Giles et al., 5 Sep 2025).

4. Ergodicity and the Environment Process

A pivotal conceptual device is the "environment seen from the particle," i.e., recentering the evolving occupation profile at the current position: $$\eta_t(x) = \eta_0(x + X_t) + \int_0^t V(x + X_t - X_s)\,ds$$ This process becomes Markovian (even when $X_t$ itself is not) and admits a stationary ergodic measure, typically a massless Gaussian free field with covariance determined by $V$ (Horvath et al., 2010). The symmetry, translation invariance, and ergodicity properties are crucial for invoking central limit theorems via martingale/Fock space and for controlling the additive functional structure.

In the critical dimension ($d=2$), the scaling limit of the environment process is a solution to a stochastic linear transport equation, with the limiting diffusivity capturing the cumulative effect of weak self-repulsion (Cannizzaro et al., 11 Mar 2024).

5. Variants and Extensions: Numerical, Geometric, and Active Polymer Models

Discrete-to-Continuum and Variants

• True self-avoiding walks (TSAW): Discrete-time, nearest-neighbor analogues defined by myopic preference for less-visited sites converge, in appropriate scaling and under mild assumptions, to SRBP in high dimensions and more complex processes in $d=1,2$ (Horvath et al., 2010, Marêché et al., 2023).
• Fractional Brownian polymers: Replacing Brownian motion by fractional Brownian motion (fBm) with Hurst parameter $H \neq 1/2$ permits modeling of polymers with anomalous memory, and leads to Flory-type scaling exponents for the end-to-end distance, $\nu_H(d) = \frac{2H+2}{d+2}$ (Bornales et al., 2011, Eleutério et al., 21 Aug 2024).
• Polymers on manifolds: SRBP generalizes to compact Riemannian manifolds, with stationary measures given by product measures (uniform over manifold, Gaussian over finite-dimensional projections), and exponential ergodicity (Benaïm et al., 2015).

Non-equilibrium and Active Matter Scenarios

• Active polymers with excluded volume: The interplay of self-repulsion and nonthermal noise (active drive) leads to non-monotonic swelling/compression transitions, modification of rheological properties under shear, and scaling exponents for end-to-end distance that cross over (e.g., from self-avoiding to Rouse regime) depending on Péclet number (Anand et al., 2020, Panda et al., 2023).

6. Open Problems and Physical Implications

• Exact characterization of scaling exponents in low-dimensional cases remains ongoing: in $d=2$, the exponent $\beta$ in $t(\log t)^\beta$ is supported to be $1/2$, but only rigorous lower and upper bounds exist for strong self-repulsion (Toth et al., 2010, Cannizzaro et al., 11 Mar 2024).
• Functional convergence and universality: While 1D SRBP exhibits superdiffusivity and anomalous scaling, discrete analogs may lack convergence in the Skorohod topology, highlighting subtle differences between models and the role of local time increment structure (Marêché et al., 2023, Kosygina et al., 2022).
• Excluded-volume effects: Comparisons among hard-core (Edwards/step function), soft-core (Gaussian Gibbs factor), and continuous self-repulsion show that the macroscopic swelling exponent depends sensitively on the form and strength of the interaction, as well as the underlying driving process (Brownian or fractional) (Eleutério et al., 21 Aug 2024).
• Complex topologies and star polymers: In star polymer models, the effective radius in 2D scales as $T^{3/4}$ (up to logarithmic factors), matching the conjectured exponent for SAW end-to-end distance, further supported by rigorous bounds (Mueller et al., 2023).

7. Summary Table: Scaling Exponents and Key Regimes

Dimension / Model	Mean-Square Displacement (MSD)	Key Scaling Exponent(s)	Reference
$d \geq 3$ (SRBP)	$\mathbb{E}|X_t|^2 \sim \sigma^2 t$	Diffusive (1)	(Horvath et al., 2010)
$d=2$ (SRBP)	$$t \log \log t \lesssim \mathbb{E}|X_t|^2 \lesssim t \log t$$, expected $t (\log t)^{1/2}$	$1 < \beta \leq 1 + \alpha$	(Toth et al., 2010, Cannizzaro et al., 11 Mar 2024)
$d=1$ (SRBP)	$$t^{5/4} \lesssim \mathbb{E}|X_t|^2 \lesssim t^{3/2}$$	Superdiffusive	(Giles et al., 5 Sep 2025)
Star polymer, $d=2$	$R_T \propto T^{3/4}$ (radius)	$3/4$	(Mueller et al., 2023)
Fractional Brownian	$R \sim N^{\nu_H}$, $\nu_H = (2H+2)/(d+2)$	$\nu_H$	(Bornales et al., 2011)

References to Related Works

• Amit, Parisi, Peliti: Origin of true self-avoiding walks (Horvath et al., 2010)
• Tóth, Werner: True self-repelling motion, conditional law uniformity (Dumaz, 2012)
• Horváth, Tóth, Vető: CLT for higher-dimensional self-repelling diffusions (Horvath et al., 2010)
• Portmann, Weber (2025): Construction and energy solutions for 1d SRBP (Giles et al., 5 Sep 2025)
• Gubinelli, Lohmann, et al.: Invariance principles and scaling limits in $d=2$ (Cannizzaro et al., 11 Mar 2024)
• van der Hofstad, van Ginkel (2023): Self-repelling star polymers, dimension dependence (Mueller et al., 2023)

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The self-repelling Brownian polymer and its numerous variants provide a rich interplay between analytic, probabilistic, and physical methodologies. They illuminate critical aspects of macromolecular statistics, anomalous diffusion, and the effect of pathwise memory in stochastic dynamics. The toolbox for their modern analysis includes singular stochastic PDEs, spectral and martingale theory, structural Fock space methods, and large-scale computational simulation.