Bold Step Acceptance Mechanism

• Bold Step Acceptance is a probability mechanism where a one-dimensional random walker makes record-dependent moves based on a gamma-controlled function p(y).
• The mechanism delineates sub-diffusive, diffusive, and ballistic regimes by adjusting the bias at the record, thereby modulating the diffusion exponent.
• Cycle decomposition analysis links local record decisions to the overall anomalous transport properties without requiring additional noise or degrees of freedom.

Bold step acceptance refers to the probability mechanism that governs the forward progression of a one-dimensional random walker at the record distance from its origin. In the bold/timorous random walk (BTRW), as formalized by Serva, the walker’s movement diverges from the simple symmetric random walk (SSRW) only when at the maximum distance ever attained from the starting point. At these “records,” the walker can either step farther (“bold”) with probability $p(M_n)$ or step back (“timorous”) with probability %%%%1%%%%, where $M_n$ denotes the running maximum. The function $p(y) = \frac{y^\gamma}{1 + y^\gamma}$, parameterized by $\gamma \in \mathbb{R}$, controls the degree of boldness and ultimately dictates the anomalous transport properties and scaling regimes of the process (Serva, 2013).

1. Formal Definition and Transition Rules

At step $n$, the walker is at position $X_n$ and the running maximum is $M_n = \max\{|X_0|, |X_1|, \ldots, |X_n|\}$. The stochastic rule for the next step ($\sigma_{n+1} = \pm 1$) depends on the relation between $|X_n|$ and $M_n$:

• If $|X_n| < M_n$ (“strictly inside the current record”), then

$$P(\sigma_{n+1} = +1) = P(\sigma_{n+1} = -1) = \frac{1}{2}.$$

• If $|X_n| = M_n$ (“on a record”), then

$$P(\sigma_{n+1} = \operatorname{sign}(X_n)) = p(M_n),$$

$$P(\sigma_{n+1} = -\operatorname{sign}(X_n)) = 1 - p(M_n),$$

where the function $p(y)$ is the bold-step acceptance law.

The canonical choice for the law on records is

$p(y) = \frac{y^\gamma}{1 + y^\gamma},$

with $\gamma \in \mathbb{R}$. Positive $\gamma$ leads to increasing boldness with depth into unexplored territory, negative $\gamma$ to greater timorousness (Serva, 2013).

2. Regimes of Bold and Timorous Behavior

The parameter $\gamma$ divides the walker’s behavior into distinct diffusion regimes:

• Bold regime: For $\gamma > 0$, $p(y) > 1/2$ for large $y$. The walker favors bold steps, resulting in persistent outward excursions.
• Timorous regime: For $\gamma < 0$, $p(y) < 1/2$ for large $y$, so the walker typically reverses at the record, resulting in hesitant, confined motion.
• Marginal case: At $\gamma = 0$, $p(y) = 1/2$, recovering the SSRW with standard diffusive behavior.

This single parameter $\gamma$ continuously interpolates between these regimes.

3. Asymptotic Scaling and Diffusion Exponents

The long-time scaling of the mean squared displacement (MSD) $\langle X_n^2 \rangle$ is governed by the exponent $\nu(\gamma)$ such that

$\langle X_n^2 \rangle \sim n^{2\nu(\gamma)}.$

The precise dependence of $\nu(\gamma)$ on $\gamma$ is a continuous, piecewise function:

$\nu(\gamma) = \begin{cases} \frac{1}{2-\gamma}, & \gamma < 0 \[1ex] \frac{1}{2-2\gamma}, & 0 \le \gamma \le \frac{1}{2} \[1ex] 1, & \gamma > \frac{1}{2} \end{cases}$

The resulting regime structure is:

$\gamma$	Diffusion regime	Scaling of $\langle X_n^2 \rangle$
$\gamma < 0$	Sub-diffusive (timorous)	$n^{2/(2-\gamma)}$ ($< n$)
$0 < \gamma < 1/2$	Super-diffusive (bold)	$n^{2/(2-2\gamma)}$ ($> n$ and $< n^2$)
$\gamma > 1/2$	Ballistic (strongly bold)	$n^2$

The crossover at $\gamma = 0$ yields normal diffusion $\langle X_n^2 \rangle \sim n$.

4. Cycle Decomposition and Mechanism

The analytical approach relies on decomposing the walk into cycles that each start with a visit to the current running maximum. Each cycle consists of:

• A “lazy excursion” of random length $m(y)$, comprising SSRW statistics inside $|X_n| < y$,
• An “active excursion” of length $n(y)$, a run of consecutive outward bold steps, extending the record.

The statistical properties are:

• For large $y$, $m(y)$ has mean $\langle m(y) \rangle \sim 2y$ and variance $\operatorname{Var} m(y) \sim \frac{8}{3} y^3$, reflecting SSRW hitting times.
• $n(y)$ has a geometric distribution with success probability $1 - p(y)$, so $\langle n(y) \rangle \sim y^\gamma$ for $\gamma \neq 0$, with variance $\sim O(y^{2\gamma})$.

Aggregating these cycles, the overall scaling of the record evolves as:

• For $\gamma < 0$, the sum of $m(y_k)$ is sharply concentrated, yielding deterministic scaling $y \sim t^{1/(2-\gamma)}$.
• For $0 < \gamma < 1/2$, heavy-tailed $m$-sums give $t \sim k^2$, $y \sim k^{1/(1-\gamma)}$, and $y \sim t^{1/(2-2\gamma)}$.
• For $\gamma > 1/2$, the duration is dominated by the record extension itself, yielding ballistic $y \sim t$.

Critically, $X_n$ and the sequence of running maxima $y(n)$ share the same scaling exponent, ensuring the validity of the MSD relation (Serva, 2013).

5. Physical Interpretation and Significance

The bold-step acceptance law $p(M_n)$ encapsulates a form of memory-dependent random walk, where the walker’s bias at the record is history dependent and dynamically adjusts with the extremity of the excursion:

• For increasing $p(y)$ (i.e., $\gamma > 0$), forward records are reinforced by streaks of consecutive outward steps, producing bursty, persistent excursions and driving the process into super-diffusive or ballistic regimes.
• For decreasing $p(y)$ ($\gamma < 0$), the walker is increasingly hesitant on the frontier, regularly retreating and sampling the surroundings, confining the trajectory and producing sub-diffusive scaling.
• The special case of $\gamma = 0$ reduces precisely to SSRW, with no memory or bias at the frontier.

Through this single mechanism, the model reproduces a continuous spectrum of anomalous transport, directly linking the local decision law at the record to macroscopic transport properties (Serva, 2013).

6. Relation to Anomalous Transport and Minimality

The bold-step acceptance framework exemplifies a minimal generative process for memory-induced anomalous transport. Without introducing additional degrees of freedom or noise structure, the $\gamma$-parameterized law $p(y)$ smoothly interpolates between distinct universality classes of random walks. The scaling exponents bridge sub-diffusive, standard diffusive, and super-diffusive (ballistic) behaviors, providing a controlled setting for the analytical study of record-dependent and reinforcement-driven random walks (Serva, 2013). This suggests a broad applicability for understanding related phenomena in non-Markovian or self-interacting random processes.