"True" self-avoiding walks on general trees

Abstract: We study the asymptotic behavior of ``true" self-avoiding random walks on general infinite locally finite trees. In this model, the walk starts at the root and, at each step, from its current vertex chooses a neighboring edge to traverse with probability proportional to the current weight of that edge, where the weight of each edge after being traversed $n$ times is given by $w(n)=\exp(-βn)$. We show that the process exhibits a sharp phase transition between recurrence and transience. The critical value is determined by the branching-ruin number of the tree, which coincides with the Hausdorff dimension of the boundary of the tree under a suitable metric. We prove that the walk is almost surely transient when the branching-ruin number is greater than $1/2$, and recurrent when it is less than $1/2$. This resolves an open question posed by Kosygina.

• The paper establishes that self-avoiding walks on trees exhibit a phase transition based on the tree’s branching-ruin number, with recurrence for values below 1/2 and transience above.
• It employs Rubin’s construction and Markov renewal theory to derive explicit asymptotic ruin probabilities, reducing the problem to one-dimensional path models.
• The findings resolve Kosygina’s conjecture and offer practical insights for applications in polymer physics, network exploration, and related fields.

Phase Transition for "True" Self-Avoiding Walks on General Trees

Introduction and Model Definition

The paper "True" self-avoiding walks on general trees (2604.24389) investigates the asymptotic recurrence and transience properties of a family of self-interacting random walks (TSAWs) on infinite, locally finite trees. The TSAW model involves a dynamic nearest-neighbor process, where transition probabilities are proportional to edge weights, which decay exponentially in the number of traversals: $w(n) = \exp(-\beta n)$. This bond-repulsion protocol induces a strong path-dependent memory, making the walk increasingly unlikely to revisit previously crossed edges.

The notion of recurrence (infinite visits to every vertex) versus transience (only finitely many visits per vertex) is analyzed with respect to the tree's combinatorial structure, specifically its branching-ruin number $\text{br}_r(\mathcal{T})$, which quantifies polynomial growth via cutsets and encodes the Hausdorff dimension of the tree's boundary.

Main Results: Phase Transition Controlled by Branching-Ruin Number

The central claim is a sharp dichotomy: TSAWs on general trees exhibit a phase transition between recurrence and transience, governed strictly by $\text{br}_r(\mathcal{T})$:

• The TSAW is almost surely recurrent if $\text{br}_r(\mathcal{T}) < 1/2$.
• The TSAW is almost surely transient if $\text{br}_r(\mathcal{T}) > 1/2$.

The threshold $1/2$ is exact and resolves a conjecture posed by Kosygina regarding the recurrence/transience behavior of self-avoiding random walks on trees of polynomial growth.

Methodological Framework

Rubin's Construction and Reduction to Path Models

The TSAW is constructed using a family of independent exponential clocks (Rubin's construction), allowing for strong coupling across subgraph restrictions and extensions. This enables the problem to be reduced to analysis of the model along geodesic paths—essentially one-dimensional settings—facilitating explicit computation of ruin probabilities and return time distributions. The restriction principle ensures that one-dimensional TSAWs encode the behavior of the full process on the tree when conditioned on infinite visitation.

Markov Chain Representation and Ruin Probabilities

A key technical component is the formulation of a Markov chain $Y$ tracking the number of backward steps before hitting a leaf in the path extension. This chain features transition kernels derived from the stationary law of another chain $\eta$, with increments governed by memory-dependent probabilities. The ruin probability $r_n$, quantifying the likelihood of hitting the endpoint before returning to the root in a path of length $n$, is shown to decay as $\text{br}_r(\mathcal{T})$0, with explicit asymptotic formulas derived via Markov renewal theory and comparison to symmetric random walk kernels.

Quasi-Independent Percolation and Cluster Analysis

The transience/recurrence dichotomy is established through the construction of a percolation model on the tree: edges are declared "open" or "closed" depending on whether the TSAW's extension process reaches the corresponding vertex before returning to the root. This induces a correlated percolation, whose quasi-independence is proven via control over the joint occurrence probability of edge events, leveraging the exact asymptotics derived for ruin probabilities and excursion events. The rigorous comparison to the classic gambler's ruin problem is circumvented with detailed analysis of memory effects in TSAW dynamics.

Cluster properties are then related to the branching-ruin number via adapted conductances and max-flow min-cut arguments; recurrence corresponds to finite open clusters almost surely (vanishing probability for infinite clusters), while transience is associated with the existence of a nonzero flow satisfying summability criteria imposed by the adapted conductance asymptotics.

Numerical Results and Contradictory Claims

The explicit result that the critical threshold for the phase transition is exactly $\text{br}_r(\mathcal{T})$1 in terms of the branching-ruin number is both strong and contrary to prior conjectures or expectations from analogous models with weaker memory (such as once-reinforced random walks or random walks in random environments). The exact asymptotic formula for the ruin probability is derived and shown to match polynomial decay with exponent $\text{br}_r(\mathcal{T})$2, providing quantitative insight that is robust across general tree architectures.

Implications and Extensions

Theoretical Implications

The resolution of Kosygina's conjecture and the introduction of a phase transition governed by the branching-ruin number place TSAWs at a critical juncture in the study of self-interacting particle systems on general graphs. The approach extends the analytical toolkit previously used for integer lattices and provides a framework for studying systems with strong non-reversibility and path-dependent dynamics.

The characterization of TSAWs as null recurrent/transient based on the tree's fractal dimension (boundary Hausdorff dimension) offers precise criteria for the existence of persistent exploration or escape dynamics in networked environments.

Practical Applications

TSAW-type processes are central to applications in polymer physics (as dynamic models for polymer growth with site/bond repulsion), network exploration algorithms, and models of biological navigation (chemotactic motion). The exact phase transition criterion can inform the design of robust exploration strategies and enhance the understanding of transport phenomena in heterogeneous, hierarchical networks.

Speculation on Future Research Directions

Future work may generalize the analysis to more complex classes of graphs, including those with random and quenched environmental disorder or higher-order memory dependence. The restriction and excursion kernel framework appears adaptable to other self-interacting models, such as excited random walks, event-chain Monte Carlo algorithms, or quantum variants. The extension to time-continuous or hybrid models may also yield further insights into scaling limits and universality classes.

Investigation into the renormalization and scaling exponents in higher-dimensional analogues or graphs with distinct spectral properties could enrich the theory and connect with research on reinforced and excited random walks, as well as percolation in complex networks.

Conclusion

This work rigorously proves a sharp phase transition for "true" self-avoiding walks on general trees, dictated by the branching-ruin number. The synthesis of Rubin's construction, Markov renewal theory, and quasi-independent percolation analysis enables precise characterization of recurrence/transience. The asymptotic formulas derived and the resolution of open questions inform the understanding of long-range memory effects in self-interacting walks and provide a template for future studies in statistical mechanics, probability, and complex networks.