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Random walks in Dirichlet random environment in dimension $d+1$

This paper numerically investigates random walks in a Dirichlet random environment across dimensions $d=1, 2, and 3$, verifying KPZ-like growth in lower dimensions, confirming a phase transition in $d=3$ , and establishing exact relationships between sample-to-sample variances and extreme diffusion coefficients.

Original authors: Guillaume Barraquand, Alexander K. Hartmann, Pierre Le Doussal

Published 2026-07-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Guillaume Barraquand, Alexander K. Hartmann, Pierre Le Doussal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, invisible forest where the rules of walking change every single second. In this forest, a traveler doesn't just wander randomly; they are guided by a map that is being rewritten by a mischievous, invisible hand at every step. This is the world of "random walks in random environments," a playground for physicists trying to understand how disorder shapes movement. Usually, if you walk randomly for a long time, you end up a certain distance away, like a drunkard stumbling home. But sometimes, the traveler takes a wild, atypical path, sprinting far beyond where they should be. Scientists have long suspected that these rare, super-fast paths behave exactly like a growing, crinkly surface—think of a crumpled piece of paper or a spreading stain—described by a famous mathematical rule called the KPZ equation. While we know this works perfectly in a one-dimensional world (a single line), the big question is: does this magic connection hold up in our real, three-dimensional world?

This paper dives into that question by simulating a specific type of walker in a forest where the "rules of the road" are chosen from a special mathematical family called the Dirichlet distribution. Think of this distribution as a fair, yet chaotic, way of assigning probabilities: at every intersection, the walker must choose a direction, and the chances of going left, right, forward, or backward are drawn from a bag of numbers that always add up to one. The researchers used powerful computers to track millions of these walkers in 1D, 2D, and 3D environments to see if their "atypical" journeys still follow the KPZ growth rules.

The results are a mix of confirmation and new mysteries. In one and two dimensions, the simulation confirms the hunch: the fluctuations of these rare paths grow in time exactly as the KPZ theory predicts, following a specific power law. It's as if the crinkly surface analogy holds true even in a flat plane. However, the story gets fascinating in three dimensions. Here, the researchers found a "phase transition," a tipping point in the walker's behavior. If the walker tries to sprint at a shallow angle (close to the average path), the chaos of the environment keeps them in check, and their path remains relatively calm. But if they aim for a steeper, more extreme angle, the disorder takes over, and their path begins to fluctuate wildly, growing in a way that suggests a new, turbulent regime.

The team didn't just guess this; they calculated an exact mathematical lower bound for where this transition happens. They found that for their specific model, the transition occurs when the walker's angle corresponds to a specific parameter value around 11.1. Below this threshold, the system is in a "weak disorder" phase where the walker's path is stable. Above it, the system enters a "strong disorder" phase where the path becomes heavy-tailed and unpredictable. They also discovered that in this chaotic 3D regime, the probability of finding the walker in a specific spot develops a "heavy tail," meaning extreme events are much more likely than in a normal, calm world.

Finally, the paper connects these wild fluctuations to a concept called the "extreme diffusion coefficient." Imagine measuring how much the average position of a crowd of walkers varies from one forest to another. The authors found that this variation is directly linked to the critical angle where the chaos begins. In simpler terms, the way the crowd's average position jitters tells you exactly when the environment becomes too wild to control. While the simulations in 3D were limited by the sheer computational power required (the calculations grow incredibly fast as time increases), the evidence strongly supports the existence of this transition and the shift from calm to chaotic behavior as the walker's ambition increases.

Technical Summary: Random Walks in Dirichlet Random Environment in Dimension $d+1$

Problem Statement
The paper investigates the atypical behavior of random walks in time-dependent random environments (RWRE), specifically focusing on the connection between these walks and Kardar-Parisi-Zhang (KPZ) growth universality. While the relationship between RWRE and KPZ growth is well-established in spatial dimension $d=1$ (where atypical trajectories exhibit Tracy-Widom fluctuations), the behavior in higher dimensions ( $d > 1$ ) remains less understood. The authors aim to numerically test theoretical predictions regarding the scaling of probability fluctuations, the existence of phase transitions between weak and strong disorder, and the nature of intermediate deviation regimes in dimensions $d=1, 2,$ and $3$.

Methodology
The study employs a discrete model of random walks on $\mathbb{Z}^{d+1}$ starting from the origin, where the transition probabilities at each space-time point are drawn independently from a Dirichlet distribution, $p \sim \text{Dir}(\alpha_1, \dots, \alpha_{d+1})$ . This model generalizes the integrable Beta RWRE known in $d=1$ .

Key methodological features include:

Explicit Stationary Measure: The authors utilize the fact that the Dirichlet RWRE admits an explicit, product-form stationary measure involving independent Gamma variables. This allows for precise analytical predictions regarding the distribution of $\log P(n)$ in typical directions.
Numerical Simulation: The authors implement a custom algorithm to simulate the evolution of probability distributions up to large times ( $t_{max}$ ). To manage memory constraints for high dimensions, they store probabilities in vectors rather than high-dimensional arrays and use a recursive approach to iterate over accessible sites on the discrete simplex.
Observables: The primary observables are the variance and higher cumulants of $\log P(X(t) = tv)$ for various direction vectors $u$ (where $v = tu$). They also compute the sample-to-sample variance of the thermal average position $\langle X(t) \rangle$ and the second moment of the ratio of probabilities $Z_t$ .
Analytical Tools: The paper derives exact expressions for the second moment of $Z_t$ using a two-replica calculation and connects the variance of the thermal average to the extreme diffusion coefficient.

Key Contributions and Results

Verification of KPZ Scaling in $d=1$ and $d=2$ :
- Typical Directions: In the diagonal direction (typical drift), the variance of $\log P(n)$ converges to a constant determined by the stationary measure, consistent with theoretical predictions involving the digamma function.
- Atypical Directions: For off-diagonal directions, the variance of $\log P(n)$ $lo g P (n)$ grows as a power law $t^{2\beta}$ $t^{2 β}$ .
  - In $d=1$ , the exponent is $\beta = 1/3$ , confirming KPZ universality.
  - In $d=2$ , the exponent is $\beta \approx 0.24$ , consistent with numerical estimates for $2+1$ dimensional KPZ growth.
- Boundary Cases: When the walk is confined to lower-dimensional subspaces (edges), the scaling follows the KPZ exponent of the reduced dimension.
Phase Transition in $d=3$ :
- The authors confirm the existence of a phase transition in $d=3$ dependent on the angle between the trajectory and the diagonal.
- Weak Disorder Phase: For angles close to the diagonal (small deviation), the variance and third cumulant of $\log P(n)$ saturate to finite constants at large times.
- Strong Disorder Phase: For larger angles, these cumulants grow with time, indicating a transition to the strong disorder regime.
- Critical Velocity: By calculating the second moment of the probability ratio $Z_t$ exactly, the authors establish a lower bound for the critical velocity $v_c \geq 0.639$ (corresponding to a critical parameter $k_c \geq 11.1$ ). Below this threshold, the second moment remains bounded; above it, the distribution of $Z_t$ develops a heavy tail.
Distributional Properties:
- In the weak disorder phase of $d=3$ , the point-to-point probability $P(X(t)=tv)$ is shown to acquire a heavy-tailed distribution (power law $P(Z) \sim Z^{-(1+\mu)}$ ).
- In the strong disorder phase, the cumulants of $\log P$ grow with time, and the skewness becomes positive, consistent with KPZ class behavior.
Thermal Average and Extreme Diffusion:
- The paper computes the sample-to-sample variance of the thermal average position $\langle X(t) \rangle^2$ .
- Analytical and numerical results show that this variance scales as $t^{1/2}$ in $d=1$ , $\log t$ in $d=2$ , and converges to a constant in $d=3$ .
- This observable is explicitly linked to the "extreme diffusion coefficient" ( $\lambda_{ext}$ ), providing a physical interpretation for the critical scale of moderate deviations.

Significance and Claims
The paper claims to provide a comprehensive numerical verification of the connection between RWRE and KPZ universality across dimensions. Its primary significance lies in:

Generalization: Extending the understanding of the Beta RWRE (integrable in $d=1$ ) to higher dimensions using the Dirichlet model, which retains the advantage of an explicit stationary measure.
Phase Transition Confirmation: Providing numerical evidence for the predicted phase transition in $d=3$ between weak and strong disorder regimes, a phenomenon analogous to directed polymers.
Exact Bounds: Deriving an exact lower bound for the critical velocity in $d=3$ via second-moment calculations.
Unified View: Demonstrating that the variance of the thermal average position serves as a predictor for the critical/moderate deviation scale in all dimensions studied.

The authors remain modest regarding the precise determination of critical exponents in $d=3$ due to computational limitations ( $O(t^4)$ scaling), noting that while the existence of the transition is confirmed, the precise value of the KPZ exponent $\beta$ in the strong disorder phase requires larger time scales than currently accessible.

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