{"id":"cf86a841-a17a-40cd-be4e-3b4c7c993260","arxiv_id":"2411.13926","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For d ≥ 5 the random walk on a Poissonian field of independent random walks satisfies a strong law and large deviation bounds, and for d ≥ 9 an annealed functional central limit theorem is claimed, for every positive particle density.","lead":"Random walks on random walks are studied in high dimensions: for d ≥ 5 the walk has a positive speed and exponential large-deviation bounds, and for d ≥ 9 a Brownian scaling limit is claimed. The results are non-perturbative: they hold for any density of the Poissonian particle field, removing the large-density and drift assumptions of earlier work.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The d≥9 aFCLT rests on a summability condition that the proved logarithmic mixing bound cannot supply; Theorem 2.2b is not established as written.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: the paper's only proved mixing estimate is logarithmic, and the square-root summability needed for the d≥9 aFCLT is incompatible with that rate. My independent reading of the proof of Theorem 2.2 confirms the erroneous implication: the sentence 'when d≥9, by the bound in Theorem 3.5, it follows that Σ√φ(2t)<∞' is arithmetically false, since a logarithmic tail is never square-root summable. This is not a matter of disagreement with an external consensus or a stylistic preference; it is an internal inconsistency between a stated sufficient condition and the quantitative bound actually derived. The SLLN and large-deviation estimates (Theorem 2.1) and the weak FCLT (Theorem 2.2a, d≥5) rely only on φ(t)→0 and therefore appear to survive the concern. The central advertised Brownian scaling limit for d≥9, however, is unsupported unless either the mixing bound is substantially improved or the companion paper [5] is shown to contain a weaker sufficient condition that logarithmic decay satisfies. The paper's own Remark 3.1 concedes the decay rate is not thought to be optimal, but no improved bound for the random-walk mixing quantity φ(t) is proved; Corollary 5.3 gives a polynomial rate only along a fixed bi-infinite path and does not control the random path of the walk. Thus the appropriate disposition matches the reader's CONDITIONAL verdict: the non-perturbative SLLN/LDB contributions stand, while the d≥9 aFCLT is unproven pending a stronger mixing estimate or a revised transfer argument. No change to the reader's verdict is needed.","tokens_in":26705,"tokens_out":3594,"duration_ms":37043,"concrete_test":"Fix d=9, take φ(t)=C (log t)^{-(d/2+2)} as allowed by Theorem 3.5, and evaluate S=Σ_{t=1}^{∞} √φ(2t). The integral test shows S diverges because ∫_2^∞ dx/(log x)^{d/4+1} = ∞. To settle whether the gap is fatal, inspect [5, Proposition 3.4] and its proof: determine whether the hypothesis Σ√φ(2t)<∞ can be weakened to a condition satisfied by logarithmic decay (e.g., φ(t)→0 plus a slowly-varying variance normalization). If no weaker condition exists, Theorem 2.2b must be re-proved with a polynomial (or faster) decay bound for φ(t), or the claim should be retracted to the waFCLT only.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.2b is derived from Proposition 3.4ii, which requires Σ_{t≥1} √φ(2t) < ∞. The only rate proved for the mixing coefficient is Theorem 3.5: φ(t) ≤ C (log t)^{-p} with p = d/2+2 (for d=9, p=6.5). Since Σ_{t≥1} (log t)^{-p/2} diverges for every p>0, the assertion in the proof of Theorem 2.2 that 'when d≥9, by the bound in Theorem 3.5, it follows that Σ_{t≥1} √φ(2t) < ∞' is false. The logarithmic decay is far too slow for square-root summability, so the annealed functional central limit theorem — the paper's headline d≥9 scaling result — does not follow from the estimates proved here. A secondary, related inconsistency: the exponent stated in Theorem 3.5 is −d/2−2, while Lemma 5.2 and the proof of Theorem 3.5 yield −d/2+2; regardless of which exponent is correct, neither gives a summable √φ-series. The weaker Theorem 2.2a for d≥5 only needs φ(t)→0 and is not affected, and the SLLN/LDB results of Theorem 2.1 also appear sound, but the central probabilistic scaling limit advertised in the abstract and introduction is left without a valid proof unless a stronger mixing estimate or a different transfer principle is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper analyzes the random walk on random walks (RWRW) model on Z^d, in which the dynamic environment is a Poissonian field of independent finite-range simple random walks of density λ, and the walker X_t makes a jump with law α(0,·) or α(1,·) according as its current location is vacant or occupied. The main claims are: for d ≥ 5 and every λ > 0, a strong law of large numbers and exponential large deviation bounds for X_t/t (Theorem 2.1); for d ≥ 5, a weak annealed functional CLT with slowly varying normalization (Theorem 2.2a, first part); and, for d ≥ 9, a genuine annealed FCLT with positive definite covariance matrix for truly d-dimensional walks (Theorem 2.2b), together with a constant normalization in the second part of Theorem 2.2a. The proof strategy is to estimate a uniform mixing coefficient φ(t) for the local environment process via a new decomposition of the conditioned Poisson environment (Theorem 4.2, developed in Section 6), obtaining φ(t) ≤ C(log t)^{−d/2−2} (Theorem 3.5), and then to import limit theorems from the authors' companion paper [5, Proposition 3.4], whose hypotheses are φ(t) → 0 (plus strong ellipticity) for the SLLN/LDB and weak FCLT, and Σ√φ(2t) < ∞ for the Brownian scaling limit.","tokens_in":26952,"tokens_out":23278,"duration_ms":201825,"significance":"Conditional on the limit theorems being correct, the paper is a significant advance: it removes the small- or large-density restrictions of Blondel, Hilário, dos Santos, Sidoravicius and Teixeira (2019), extends the SLLN/LDB to all densities in d ≥ 5 with exponential rather than stretched-exponential deviations, and provides the first high-dimensional Brownian scaling result for this model at arbitrary density. The strength of the paper is that the mixing estimate is derived from first principles: the Poisson decomposition of Section 6 and the domination Theorem 4.2 are original tools, applied to classical heat-kernel bounds, with no fitting parameters or definitional circularity; the SLLN/LDB and the slowly varying waFCLT appear sound on my reading. The weakness is that the stress-test concern lands: the d ≥ 9 conclusions (Theorem 2.2b and the constant-σ part of Theorem 2.2a) are derived from a false summability implication, and the exponent in Theorem 3.5 is not the exponent proved in Lemma 5.2. The advertised Brownian scaling results therefore do not follow from the estimates supplied in the manuscript.","major_comments":[{"comment":"The proof asserts that 'when d ≥ 9, by the bound in Theorem 3.5, it follows that Σ_{t≥1} √φ(2t) < ∞', and this summability is the only route used to reach Theorem 2.2b and the constant-σ part of Theorem 2.2a via Proposition 3.4. The implication is false: the estimate φ(2t) ≤ C(log(2t))^{−d/2−2} is compatible with, for example, φ(2t) = C(log(2t))^{−d/2−2}, for which √φ(2t) ≥ c (log t)^{−(d/4+1)}, and Σ_{t≥1} (log t)^{−p} diverges for every p > 0 (by Cauchy condensation, the condensed series is Σ_k 2^k (k log 2)^{−p} = ∞). Hence the bound in Theorem 3.5 does not verify the summability hypothesis of Proposition 3.4i–ii regardless of how the exponent is read, and Theorem 2.2b together with the h(n) = σ assertion of Theorem 2.2a are not established by the submitted argument. Theorem 2.1 and the slowly varying part of Theorem 2.2a are unaffected, since they require only φ(t) → 0, which both exponents supply. Repair requires either a genuinely faster decay of φ(t) (of order t^{−(2+ε)} would suffice for square-root summability) or a transfer principle from [5] that does not demand Σ√φ(2t) < ∞.","section":"Section 3, Proof of Theorem 2.2"},{"comment":"The exponent in Theorem 3.5 does not match the exponent that the proof produces. Theorem 3.5 states φ(t) ≤ C(log t)^{−d/2−2}, while Lemma 5.2 states the bound Cn^{−d/2+2}, and the proof of Theorem 3.5 substitutes (T')^{−d/2+2} with T' = C log T, obtaining (log T)^{−d/2+2}; the two exponents differ by 4. As written, the stated theorem is strictly stronger than anything Lemma 5.2 and the proof of Theorem 3.5 supply, and Remark 3.1 compounds the confusion by describing Corollary 5.3 (exponent −d/2+2) as an improvement over Theorem 3.5. The authors should align the statement of Theorem 3.5, Lemma 5.2, and the proof. Note that this alignment does not fix the gap in Major Comment 1, since both candidate rates are logarithmic and neither implies Σ√φ(2t) < ∞.","section":"Section 3, Theorem 3.5, and Section 5, Lemma 5.2"},{"comment":"The statement of the central decomposition theorem is not well-formed as printed. The display 'P^{(1)}_{I(1)}(·|C^{(1)}) is equal, in law, to Σ_y P^{(1)}_{I(1)}(y|C^{(1)}) Σ_{k=1}^{κ} L(Y^{(k)}|y)' mixes the weights P^{(1)}_{I(1)}(y|C^{(1)}) with an unnormalized sum of marginal laws L(Y^{(k)}|y), so the right-hand side is not readable as a probability measure; the fourth bullet, asserting independence of the Y^{(k)} given y, suggests that the intended object is the joint law of (Y^{(1)},...,Y^{(κ)}) given y, but the sum of marginals does not say this. The first bullet defines y = (y_1,...,y_K) with K ≤ κ, while the sums run to κ, and the third bullet introduces laws L(Y^{(k)}|y) for k > K that do not occur in the display; the stochastic domination sentence in the same theorem similarly writes Σ_{k=1}^{|y|} instead of κ. Since Theorem 6.6 is the basis of Theorem 4.2 and hence of the ellipticity and mixing estimates used throughout the paper, this statement needs a careful rewrite (mixture weights, joint law of the extra paths, and domination) before the proof of Theorem 4.2 can be checked.","section":"Section 6.5, Theorem 6.6"}],"minor_comments":[{"comment":"The cross-references to the ellipticity and diverging-variance results are inconsistent with their statements: Section 3 ends with 'the proof of Proposition 3.3', but the result is stated as Lemma 3.3, and Section 4.2 contains 'Proof of Proposition 3.1' for a result stated as Lemma 3.1.","section":"Sections 3 and 4.2"},{"comment":"In the proof of Lemma 3.3, Eq. (3.2) contains a typo, 'Pω(‖Xn − E(Xn)‖)2 > ǫ)': it should read Pω(‖Xn − E(Xn)‖₂ > ǫ); the notation W^{(j)}_{[1,i_j]} later in the same proof also uses inconsistent subscripts.","section":"Section 3, Eq. (3.2)"},{"comment":"In the proof of Lemma 5.2, the event {τ_m < ∞} is conditioned on Q[0,T] ≥ 1, but T is not defined at that point (the lemma's parameter is n); the closing sentence of the proof also refers to 'Proposition 5.2', which should be Lemma 5.2.","section":"Section 5, proof of Lemma 5.2"},{"comment":"The products in (4.7)–(4.8) are indexed inconsistently: the factors run over s = 1,...,R−1 and s = R,...,|γ|, while the preceding lines index particles by i with anchor points z_i = −s; rewriting everything in terms of the anchors z_i would remove the ambiguity, and the symbol '∩' in the middle of (4.6) should be a multiplication sign.","section":"Section 4.2, Eqs. (4.6)–(4.8)"},{"comment":"The main theorems are stated for λ ∈ [0,∞), but the supporting lemmas (Lemma 3.1, Theorem 3.5, Proposition 3.4) assume λ > 0; the degenerate case λ = 0 reduces to a homogeneous random walk with kernel α(0,·) and is trivial, but the papers should say so explicitly.","section":"Section 2, Theorems 2.1 and 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's transfer results are quoted from the authors' own companion preprint [5] (arXiv:2303.06756), which has not yet been peer-reviewed, and the present manuscript fails to verify the summability hypothesis it imports; I would recommend that the two papers be evaluated jointly and that [5, Proposition 2.1] be checked for any sharper hypotheses. The environment decomposition of Section 6 is the most valuable original contribution and deserves careful editorial scrutiny, because its key statement, Theorem 6.6, is currently very hard to parse and its notation (κ versus K, sums of marginals versus joint laws) must be corrected before the proof of Theorem 4.2 can be verified. If the summability gap cannot be closed, the authors should weaken the statements of Theorems 2.2a (second part) and 2.2b accordingly, rather than leaving the false implication in the proof of Theorem 2.2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the headline d≥9 annealed FCLT (Theorem 2.2b, and the constant-variance claim in 2.2a) is not established by the paper's own estimates. The proof reduces it to Σ√φ(2t)<∞, imported from the companion paper [5]. The only rate proved here is φ(t) ≤ C (log t)^{-p} (with an exponent that is −d/2−2 in the theorem statement but −d/2+2 in the proof). For d=9 that's at best (log t)^{-2.5}, and Σ (log t)^{-p/2} diverges for every p>0. So the asserted implication is false as written. That's a load-bearing gap, not a cosmetic one: the Brownian scaling limit is the central advertised result.\n\nWhat's actually good: for d≥5, Theorem 2.1 (SLLN and exponential LDBs for every λ>0) follows from the proved φ(t)→0 and appears sound. The environment decomposition in Section 6, especially the domination theorem for conditional Poisson point processes (Theorem 4.2/6.6), is a genuinely new and substantial technical tool, and the proof is mostly self-contained. The removal of the perturbative assumptions (large density/drift) from [6] is a real advance if the SLLN/LDB part survives refereeing.\n\nSoft spots, in proportion: (1) the FCLT gap above; (2) an internal exponent mismatch — Theorem 3.5 says −d/2−2, but Lemma 5.2 and the proof of Theorem 3.5 give −d/2+2. Whichever is correct, it does not save summability, but it needs fixing either way; (3) the limit theorems quote the authors' unpublished companion [5] as a black box. That's acceptable if [5] is sound, but a referee cannot fully verify without it, so the authors should either include the needed statements with proofs or clearly state which parts are conditional on [5].\n\nThis paper deserves a serious referee. The SLLN/LDB results and the domination tool are worth publishing even if the FCLT has to be cut or restricted. I'd send it to review, with instructions to focus on the summability step and the companion-paper dependence.","headline":"The d≥9 FCLT is not established: the proved logarithmic mixing rate is too slow for the Σ√φ(2t)<∞ premise, and the theorem statement and proof disagree on the exponent; the SLLN/LDB for d≥5 look sound.","tokens_in":27572,"tokens_out":2685,"would_cite":false,"duration_ms":25116,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K37","60F17","60G55","60J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that in dimensions five and higher, a walker moving through a Poissonian swarm of independent particles travels at a deterministic linear speed, and in dimension nine and higher the walk converges to Brownian motion.","keywords":["random walk in dynamic random environment","Poissonian field of random walks","strong law of large numbers","large deviations","functional central limit theorem","uniform mixing","annealed law","high dimensions"],"falsifier":"One concrete check is to compute $S_d=\\sum_{t\\ge1}\\sqrt{\\varphi(2t)}$ under the paper's stated bound: since $\\varphi(2t)\\le C(\\log t)^{-d/2-2}$, the terms are $C(\\log t)^{-(d/2+2)/2}$, whose sum diverges for every $d$; this means the stated derivation of Theorem 2.2b requires a strictly stronger mixing estimate than Theorem 3.5 provides.","tokens_in":26447,"feed_emoji":"🎲","tokens_out":11881,"duration_ms":108177,"temperature":0.7,"pith_summary":"Consider a single walker on $\\mathbb{Z}^d$ whose next jump depends on whether a Poissonian cloud of independent random-walking particles currently occupies its site. The paper claims that for $d\\ge 5$ the walker's displacement satisfies a strong law of large numbers with a deterministic velocity, and that deviations from that velocity are exponentially rare, for every positive particle density $\\lambda$. For $d\\ge 9$ and genuinely $d$-dimensional finite-range jump kernels, the claim is stronger: the centered path converges under the annealed law to a Brownian motion with a positive-definite covariance matrix. The results are non-perturbative because no assumption on the density or on a drift is needed, whereas earlier work required large (or small) $\\lambda$ and a drift condition. This matters because the model is a testing ground for random walks in slowly mixing dynamic environments, where low-dimensional heuristics predict sub- or super-diffusive behaviour.","feed_headline":"In d≥9, a walk through moving particles scales to Brownian motion","feed_subtitle":"The results hold at every positive particle density, ending the perturbative-regime assumption in earlier work.","key_machinery":"The central object is the local environment process $(\\xi_t)=(\\omega_t(X_t), X_{t+1}-X_t)$, recording the environment bit and jump at the walker's current location, together with its uniform mixing coefficient $\\varphi(t)$. The coefficient is the largest total-variation distance, over two arbitrary conditioned past observations, between the laws of the future tail of $(\\xi_t)$. The key estimate, Theorem 3.5, bounds $\\varphi(t)$ by $C(\\log t)^{-d/2-2}$. The proof of that estimate rests on Theorem 4.2, a decomposition of the conditioned Poissonian environment: conditionally on any past observation, the field is, in law, the unconditioned field plus extra random-walk trajectories anchored to the observed path, with a stochastically dominated number of extra particles per anchor and heat-kernel decay bounds on their future influence. Strong ellipticity and diverging variance of the walk are then derived from the same decomposition.","core_discovery":"The central discovery is that the random walk on random walks is diffusive in high dimensions at every positive density. The paper proves an SLLN and exponential large deviation bounds for $d\\ge 5$ (Theorem 2.1), a weak annealed functional central limit theorem for $d\\ge 5$ with a slowly varying normalization (Theorem 2.2a), and, for $d\\ge 9$ with truly $d$-dimensional finite-range jump kernels, a full annealed functional central limit theorem with constant positive covariance matrix (Theorem 2.2b). The load-bearing estimate is the uniform mixing bound $\\varphi(t)\\le C(\\log t)^{-d/2-2}$ of Theorem 3.5: conditioned on any two possible past observations of the environment along admissible walk trajectories, the laws of the future local environment differ by at most this amount in total variation. Combined with strong ellipticity and diverging variance, that bound feeds the general limit-law machinery of the companion paper [5] to produce the theorems. The paper also contrasts the results with low dimensions: for $d\\le 2$, occupation-time tails of the environment decay slower than exponentially, so the exponential large deviation bound cannot hold there.","pith_inferences":["Editorial inference (not a paper claim): the d≥9 Brownian conclusion is derived through the summability condition $\\sum_{t\\ge1}\\sqrt{\\varphi(2t)}<\\infty$, but inserting the paper's own bound $\\varphi(2t)\\le C(\\log t)^{-d/2-2}$ gives a divergent series; the theorem as stated therefore needs a sharper mixing estimate or a different argument.","Editorial inference: the Section 6 decomposition of conditioned Poisson point processes is developed in a general setting and should transfer to other dynamic environments built from independent transient particles with finite-range motion.","Editorial inference: the authors expect the logarithmic rate to be non-optimal; if a polynomial decay with exponent greater than 2 could be proved, the same machinery would give the Brownian limit at lower dimensions than the d≥9 threshold.","Editorial inference: a concrete high-value test is to run the model at $d=4$ and $d=5$ with non-symmetric finite-range kernels and measure the velocity and variance; the paper predicts ballistic and weakly diffusive behaviour in those regimes."],"forward_implications":["For $d\\ge 5$ and any $\\lambda>0$, the walker has a deterministic asymptotic velocity, so arbitrarily sparse moving environments do not induce sub-ballistic motion.","Exponential large deviation bounds replace the stretched-exponential bounds of the earlier high-dimensional results wherever the models overlap.","For $d\\ge 9$ and truly $d$-dimensional kernels, the annealed walk converges to Brownian motion with positive-definite covariance, so the long-time spread grows as $\\sqrt{n}$ in every direction.","For $d\\ge 5$, one-dimensional projections whose variance diverges satisfy a weak annealed functional central limit theorem, with a slowly varying normalization.","The reduction to a uniform mixing estimate means any later improvement in $\\varphi(t)$ automatically sharpens the limit theorems."],"supporting_citations":[{"why":"Supplies the general limit-law machinery: once $\\varphi(t)$ decays or is summable, the SLLN, large deviations, and functional central limit theorems for the walk follow.","marker":"[5]"},{"why":"Establishes the previous high-dimensional perturbative results (large density and drift assumptions) that this paper extends, and provides the Poisson-point-process construction of the environment used here.","marker":"[6]"},{"why":"Introduces the weak uniform mixing and ellipticity tools for random walk in dynamic random environment that the present proof adapts.","marker":"[4]"},{"why":"Provides the heat-kernel estimates for simple random walks that yield transience, the positivity of the minimal density, and the bounds (4.2)--(4.3).","marker":"[19]"},{"why":"Esseen's concentration bound is invoked in Proposition 3.3 to prove the walk's variance diverges in $d\\ge 3$.","marker":"[10]"},{"why":"Shows that in $d\\le 2$ the large-deviation tails of the environment's occupation time are subexponential, providing the low-dimensional contrast that motivates the dimension threshold.","marker":"[9]"}],"fun_headline_variants":["High-dimensional walks in moving crowds become Brownian","Any density: random walks in random environments turn Gaussian in d≥9","Non-perturbative CLT for random walks in dynamic environments","For d≥9, dynamic random walks obey Brownian scaling at every density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the local environment process is uniformly mixing—given any two possible past observations, the future environment seen by the walker becomes close in total variation as the time gap grows—and, for the Brownian limit, that the decay is strong enough for $\\sum_{t\\ge1}\\sqrt{\\varphi(2t)}$ to converge; the paper proves the decay but does not show that this particular series converges from its stated logarithmic bound.","fun_headline_variants_meta":{"raw":{"variants":["High-dimensional walks in moving crowds become Brownian","Any density: random walks in random environments turn Gaussian in d≥9","Non-perturbative CLT for random walks in dynamic environments","For d≥9, dynamic random walks obey Brownian scaling at every density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000904,"raw_usage":{"total_tokens":3897,"prompt_tokens":957,"completion_tokens":2940,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":2866}},"tokens_in":573,"tokens_out":2940,"duration_ms":18073,"temperature":1.0,"reasoning_tokens":2866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:44:48.724077+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to compute $S_d=\\sum_{t\\ge1}\\sqrt{\\varphi(2t)}$ under the paper's stated bound: since $\\varphi(2t)\\le C(\\log t)^{-d/2-2}$, the terms are $C(\\log t)^{-(d/2+2)/2}$, whose sum diverges for every $d$; this means the stated derivation of Theorem 2.2b requires a strictly stronger mixing estimate than Theorem 3.5 provides.","supporting_citations":[{"cited_title":"Limit laws for random walks in a dynamic path-cone mixing random environment","cited_arxiv_id":"2303.06756","evidence_quote":"Supplies the general limit-law machinery: once $\\varphi(t)$ decays or is summable, the SLLN, large deviations, and functional central limit theorems for the walk follow."},{"cited_title":"Blondel, M","cited_arxiv_id":null,"evidence_quote":"Establishes the previous high-dimensional perturbative results (large density and drift assumptions) that this paper extends, and provides the Poisson-point-process construction of the environment used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the weak uniform mixing and ellipticity tools for random walk in dynamic random environment that the present proof adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the heat-kernel estimates for simple random walks that yield transience, the positivity of the minimal density, and the bounds (4.2)--(4.3)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Esseen's concentration bound is invoked in Proposition 3.3 to prove the walk's variance diverges in $d\\ge 3$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that in $d\\le 2$ the large-deviation tails of the environment's occupation time are subexponential, providing the low-dimensional contrast that motivates the dimension threshold."}],"review_version":1}