Ballisticity Criteria in Stochastic Processes

• Ballisticity criteria are rigorous conditions that define and quantify the transition from mere directional transience to positive-speed (ballistic) propagation in stochastic processes.
• They incorporate models such as random walks in random environments, growing graphs, and nanoelectronic transport, using methods like decay bounds, moment inequalities, and finite-volume checks.
• Establishing these criteria provides actionable insights for proving central limit theorems and understanding phase transitions in complex, disordered systems.

Ballisticity criteria formalize and quantify the distinction between mere directional transience and genuinely ballistic (i.e., positive-speed) regimes in stochastic processes such as random walks in random environments (RWRE), growing graphs, nanoscale electron transport, and impact mechanics. The mathematical and physical literature has developed a variety of rigorous criteria—expressed as decay bounds for escape probabilities, moment inequalities, finite-volume box checks, or explicit drift-to-fluctuation comparisons—as necessary or sufficient for linear-in-time escape or propagation, and for central-limit-type fluctuation results. This article surveys the principal ballisticity criteria across major models, with emphasis on high-dimensional RWRE, nonequilibrium growing graphs, kinetic transport, and material impact settings.

1. Ballisticity in Random Walks in Random Environments: Core Criteria

In multidimensional RWRE under i.i.d. uniformly elliptic environments, ballisticity criteria are centered around finitary or asymptotic bounds on backtracking (slab exit) probabilities. The classical hierarchy is as follows:

• Directional transience in direction $\ell\in S^{d-1}$:

$$P_0\left(\lim_{n\to\infty} X_n\cdot\ell = +\infty\right) = 1,$$

ensures escape but does not guarantee positive speed (Metkar et al., 11 Jan 2026).

• Ballisticity: There is $v$ with $v\cdot\ell > 0$ such that

$$\lim_{n\to\infty} \frac{X_n}{n} = v, \qquad \text{almost surely}.$$

• Sznitman's (T) condition: Exponential annealed decay of "back exit" probabilities from large slabs orthogonal to $\ell$:

$$P_0\left(T^-_L < T^+_L\right)\leq e^{-cL}, \quad L\gg1.$$

$(T)_\gamma$ with exponent $\gamma\in(0,1)$ is a stretched exponential variant (Guerra et al., 2018, Metkar et al., 11 Jan 2026).

• (T$^\prime$) condition: $(T)_\gamma$ holds for all $\gamma \in (0,1)$. It is strictly stronger and was central in Sznitman’s conjecture (Guerra et al., 2014, Guerra et al., 2018).
• Polynomial (P$_M$) condition: There is $M>0$ such that

$P_0\left(T^-_L < T^+_L\right) \leq L^{-M}$

for large $L$. This is a practical finite-volume criterion (Berger et al., 2012, Campos et al., 2012, Guerra et al., 2018).

• Effective criterion: Finitary criterion involving moments of exit ratios from well-proportioned slabs or boxes,

$$\inf_{a\in(0,1], B} L^{\alpha}\mathbb{E}[p_B^a] < 1,$$

for suitable $B$ aligned with the target direction $\ell$ (Drewitz et al., 2010, Guerra et al., 2014).

Equivalence and sufficiency: Under i.i.d. uniformly elliptic environments in $d\ge2$, the conditions (T), (T$^\prime$), $(T)_\gamma$, and sufficiently strong polynomial $(P_M)$ (with $M\ge 15d+5$) are equivalent and imply ballisticity and central limit theorems (Guerra et al., 2018, Berger et al., 2012, Campos et al., 2012). In mixing but non-i.i.d. environments, polynomial and effective criteria remain equivalent under strong enough mixing assumptions, shifting only exponents and technical constants (Guerra et al., 2019).

2. Local and Moment-Based Ballisticity Criteria

Under non-uniform ellipticity or additional structured disorder, local conditions or moment inequalities provide alternatives to slab-exit criteria:

• Ellipticity moment conditions: For $d\ge2$, ballisticity is implied if

$$\kappa(\{\alpha(e)\}) = 2\sum_{e\in U} \alpha(e) - \max_{e\in U}[\alpha(e) + \alpha(-e)] > \beta$$

with finite negative moments $$\mathbb{E}\big[\prod_{e\in U}\omega(0,e)^{-\alpha(e)}\big]$$ (Bouchet et al., 2013).

• Box-exit (B$_a^b$) criterion: There exist $R, c > 0$ such that $\mathbb{E}_0[T_{B_R}^{a+c}]<\infty$ and $R > a(a+c)/(b c) - 2$, with $T_{B_R}$ the exit time from a box (Ramírez et al., 2021).
• Drift-fluctuation (high-dimension) criterion: In $d\ge4$, under small perturbations,

$$\mathbb{E}[d(0,\omega)\cdot e_1] > C(d,q) (\mathbb{E}|\omega(0,e)-\mathbb{E}\omega(0,e)|^q)^{1/q}$$

suffices for ballisticity (Fukushima et al., 2019).

• Local trap exclusion: In 2D, the sum of negative-moment exponents in wedge- and edge-shaped local traps exceeding 1 prevents trapping and ensures ballisticity; for Dirichlet environments this gives sharp criteria (Ramírez et al., 2021).
• Non-elliptic local percolation/martingale criteria: Ballisticity can be forced by properties of the support of transition laws (no axes blocked, full spanning set available), combined with percolation thresholds or growth of range (Holmes et al., 2016).

3. Regeneration Times and Consequences

A unifying principle underlying all ballisticity criteria is control of regeneration times $\tau_k$ (random record times along direction $\ell$):

• Finiteness of $\mathbb{E}[\tau_1]$ is both necessary and sufficient (under moment regularity) for ballisticity, since it yields

$$\frac{1}{n}\sum_{k=1}^{n}(X_{\tau_k} - X_{\tau_{k-1}}) \to v, \text{ with } v\cdot\ell>0,$$

and central limit theorems follow from higher finite moments (Metkar et al., 11 Jan 2026, Campos et al., 2012, Drewitz et al., 2010).

• Slab-exit decay $\implies$ regeneration moments: Polynomial or faster decay of $q(L)=P_0[T^-_L<T^+_L]$ enforces exponential or polynomial tails for $\tau_1$, and thus nonzero limiting velocity (Berger et al., 2012, Bouchet et al., 2013).
• Sharpness: In non-uniformly elliptic or degenerate environments (e.g., only "one-point irreducibility"), failure of finite mean box-exit times is equivalent to zero velocity (Ramírez et al., 2021).

4. Technological and Physical Ballisticity Criteria

Beyond RWRE, analogous criteria govern the transition from diffusive to ballistic regimes in kinetic transport and impact models:

• Knudsen (ballisticity) number in rarefied gas and superfluid physics: The ballistic regime is defined by

$K_n = l_s/R_i > 1,$

where $l_s$ is the mean free path and $R_i$ the obstacle size, marking the onset of single-collision-dominated transport (Kleimenicheva et al., 2013).

• Ballistic transport in nano-FETs: The degree of ballisticity $R$ is defined as

$R = I_{non-ballistic}/I_{ballistic}\leq1,$

and is maximized by tuning channel geometry (diameter, length), with criteria encapsulated through mean free path to channel length ratios and voltage-division effects (Khan et al., 2018).

• Ballistic impact in composite laminates: For composite materials, the Cuniff criterion relates limit velocity $V_{50}$ to a multiscale energy parameter built from tensile strength, modulus, volume fraction, and effective sound speed:

$$V_{50} \propto \left( \frac{\sigma_c^2}{E_c} \right)^{1/3} \left( \frac{E_{bundle}}{\rho_{bundle}} \right)^{1/6},$$

providing both scaling prescription and experimental calibration (Signetti et al., 2021).

5. Ballisticity Phase Transitions and Extensions

Many systems exhibit a sharp transition between ballistic and sub-ballistic (or non-ballistic) behavior as key parameters cross thresholds:

• RWRE critical exponents: Satisfying $(P_M)$ with $M>d-1$ is now known to be the weakest polynomial decay guaranteeing the ballistic regime, resolving old conjectures (Guerra, 2020).
• Sharp thresholds in dynamic random environments: For random walks in exclusion-process environments, there exists a critical density $\rho_c$ such that for $\rho<\rho_c$ the walk is recurrent/sub-ballistic and for $\rho>\rho_c$ it is ballistic, with phase transition proved by renormalization and coupling methods (Conchon--Kerjan et al., 2024).
• Tree-builder and growing graphs: In Tree-Builder Random Walks (TBRW), the parity of the growth schedule (odd vs even $s$) is essential: odd $s$ guarantees ballisticity under a uniform ellipticity (branching) assumption, contrasting with potential trapping and recurrence for even $s$ (Iacobelli et al., 2019).
• Hierarchical/random graph traces: For biased random walks on traces of prior walks, explicit comparison of drift parameters determines the regime: if global bias $\beta$ is less than the critical $\alpha$ determined by the Laplace exponent, the walk is ballistic, else it is sub-ballistic (Croydon et al., 2019).

6. Proof Methodologies and Criterion Equivalence

The verification and theoretical development of ballisticity criteria rely on key methodological pillars:

• Multiscale renormalization: Construction of good and bad boxes, seed and recursive estimates, and block decoupling to control paths through random media (Guerra et al., 2019, Drewitz et al., 2010, Guerra et al., 2018).
• Effective criterion: Translation of asymptotic conditions into checkable finite-volume inequalities, typically involving moments or probabilities of atypical (backward) exits (Drewitz et al., 2010).
• Coupling arguments: Stochastic domination of escape probabilities, comparison with biased random walks or auxiliary processes, and use of strong Markov properties in block structures (Iacobelli et al., 2019, Guerra, 2020).
• Concentration inequalities and martingale deviation bounds: Employed especially in high-dimensional settings to upgrade local drift averages over environmental fluctuations (Fukushima et al., 2019).
• Construction/detection of local traps: For non-uniform or non-elliptic environments, geometric or probabilistic exclusion of trapping configurations ensures applicability of global criteria (Ramírez et al., 2021).

Crucial result: Under strong enough polynomial decay in slab-exit (with $M$ beyond a critical dimension-dependent threshold), all existing criteria—including (T), (T$^\prime$), effective, and polynomial—are equivalent. This equivalence has been established i.i.d. (Guerra et al., 2018, Berger et al., 2012), mixing (Guerra et al., 2019), and even under some non-uniform ellipticities (Bouchet et al., 2013).

7. Open Problems and Recent Advances

• Weakest ballisticity hypotheses: Guerra Aguilar has shown that $M>d-1$ polynomial decay is sufficient (Guerra, 2020); the natural conjecture is that mere directional transience in an open set should suffice, but no proof exists to date.
• Box-exit moment sharpness: It is proven that having finite mean exit time from some finite box is necessary and sufficient for positive speed in elliptic environments.
• Non-i.i.d., dynamically evolving, or non-reversible environments: Results such as the sharp threshold for ballisticity in particle-exclusion environments (Conchon--Kerjan et al., 2024) and renormalization in mixing RWRE (Guerra et al., 2019) extend classical theory beyond the i.i.d. static regime.
• Ballistic central limit theorems: Under all main classes of sharp ballisticity criteria, annealed and (with stronger moment controls) quenched CLTs hold (Campos et al., 2012, Guerra et al., 2019).
• Explicit constructions: Examples in Dirichlet environments, high-dimensional i.i.d. settings without Kalikow's drift field, and growing graphs provide nontrivial models satisfying ballistic but not stricter criteria (Bouchet et al., 2013, Fukushima et al., 2019, Iacobelli et al., 2019).

References and Selected Paper Overview

Main criterion/setting	Sharp condition	Key reference
Classical i.i.d. RWRE (uniform ellipticity)	(T), (T'), (P_M), effective	(Guerra et al., 2018, Berger et al., 2012, Metkar et al., 11 Jan 2026)
Weak ellipticity, local trap exclusion	(E')$_\beta$, (B)$_a^b$	(Bouchet et al., 2013, Ramírez et al., 2021)
High $d$ RWRE with disorder	$\mathbb{E}[d(0)\cdot\ell] > C\sigma_q$	(Fukushima et al., 2019)
Mixing/non-i.i.d. environments	(P$_J$), effective, (T$^\prime$)	(Guerra et al., 2019)
Dynamic exclusion environment	sharp $\rho_c$ threshold	(Conchon--Kerjan et al., 2024)
Nanoelectronic ballistic transport	$L$ vs. mean free path, $$R=I_{\mathrm{non-ballistic}}/I_{\mathrm{ballistic}}$$	(Khan et al., 2018)
Impact mechanics (composite laminates)	Multiscale $\mathcal{U}_m$ parameter	(Signetti et al., 2021)
Tree-builder/growing graphs	Parity of $s$, ellipticity of growth	(Iacobelli et al., 2019)

In summary, ballisticity criteria are quantitative, model-specific statements—typically in the form of escape, moment, or finite-volume estimates—that guarantee positive propagation speed and, under further conditions, invariance principles for complex stochastic systems. Their precise thresholds, dependence on mixing and ellipticity, and their adaptation to dynamically evolving and physically motivated settings remain active subjects of mathematical research.