Kontorovich lagarias 2009 stochastic models

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conjecture_2_1: The 3x+1 Growth Exponent Conjecture as the survey states it, credited to Applegate and Lagarias: for every integer a not divisible by 3, the count of integers |n| <= x whose 3x+1 orbit contains a is x^(1+o(1)).

conjecture_4_1: The survey's conjecture, drawn from the repeated random walk model of Lagarias and Weiss, that limsup sigma_inf(n)/log n over positive n is finite and equals the model constant gamma_RRW, about 41.677647.

theorem_6_4: The duality of Lagarias and Weiss as the survey states it: the forward repeated random walk model and the backward branching random walk models of 3x+1 iteration predict the same extremal scaled stopping constant, about 41.68.

theorem_6_5: The model analogue of the 3x+1 growth exponent conjecture, credited to Lagarias and Weiss: in the simplest backward branching random walk, the number of individuals up to the bound x grows as x^(1+o(1)) almost surely.

theorem_8_10: The survey's model prediction for the 5x+1 growth exponent: in the simplest backward branching random walk for the 5x+1 map, the number of individuals of size at most x is almost surely x^(0.650919...+o(1)), against the 3x+1 model's exponent 1.

theorem_8_3: The survey's 5x+1 forward models: the biased random walk with positive drift diverges with probability one, and so does every walk of the repeated model, a prediction the survey reads as density one divergence and turns into a warning about 3x+1 heuristics.

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Alex V. Kontorovich and Jeffrey C. Lagarias, Stochastic models for the 3x+1 and 5x+1 problems, arXiv:0910.1944v1 (2009; published in "The Ultimate Challenge: The 3x+1 Problem", AMS, 2010; 66 pp.).

Surveys and develops the repeated-random-walk and branching-random-walk models of Lagarias and Weiss for 3x+1 orbits side by side with 5x+1 models, some of them new, studied for comparison: the same heuristics predict that most 5x+1 orbits diverge, which this corpus treats as a check against arguments that would prove too much. The models admit rigorous analysis, and their behavior gives heuristic predictions, stated as conjectures, for the actual orbits; the Structure Theorems of Section 5 concern the symbolic dynamics of the accelerated map itself and are surveyed from Sinai and Kontorovich-Sinai. Most of the rigorous results it states are quoted from Lagarias and Weiss, Borovkov and Pfeifer, Sinai, Kontorovich and Sinai, and others; the new $5x+1$ results of §8 come with proofs or proof sketches (pp. 45--56). The volume also contains Lagarias's overview, which the site cites for problem 1135 as [La10]; the site's page does not cite this chapter.

Result pages:

• Conjecture 2.1 (p. 17): the $3x+1$ growth exponent $\eta_3(a)$ exists and equals $1$ for every $a\not\equiv0\pmod3$ (credited to Applegate and Lagarias).
• Conjecture 4.1 (p. 25): the $3x+1$ scaled stopping constant $\gamma=\limsup\sigma_\infty(n)/\log n$ is finite and equals $\gamma_{RRW}\approx41.677647$, the constant of Theorem 4.1 (p. 24).
• Theorem 6.4 (p. 38): the repeated random walk and branching random walk models give the same scaled stopping limit, $\gamma_{RRW}=\gamma_{BP}$ (Lagarias and Weiss).
• Theorem 6.5 (p. 39): in the $3x+1$ branching random walk $\mathcal B[1]$ the count of progeny up to $x$ is almost surely $x^{1+o(1)}$ (Lagarias and Weiss).
• Theorems 8.2 and 8.3 (pp. 46--47): every trajectory of the $5x+1$ random walk models diverges almost surely, with the survey's warning that such models cannot see a measure-zero set of divergent orbits.
• Theorem 8.10 (pp. 55--56): in the $5x+1$ branching random walk $\mathcal B[1]$ the count of progeny of size at most $x$ is almost surely $x^{\eta_{5,BP}+o(1)}$ with $\eta_{5,BP}\approx0.650919$.

Read status: claims checked for the statements on the result pages, read clause by clause on the page images of the print; proofs followed only where the survey gives them (Theorems 6.4, 8.2, 8.3 and the sketch of 8.10). Nothing here is independently reviewed.

Source: PDF. The arXiv record names arXiv's non-exclusive distribution license (arXiv:0910.1944), every other right reserved.

Bears on. #1135: the survey proves nothing about the problem, whose map is the survey's $T$ on the positive integers. Its Conjecture 4.1 (p. 25), unproved, would imply an affirmative answer with $\sigma_\infty(n)\le(\gamma_{RRW}+o(1))\log n$, by the corpus's one-line deduction on that page; an affirmative answer would give $\eta_3(1)=1$ in Conjecture 2.1 (p. 17), as the survey notes (p. 18). The rigorous results it states about the map itself concern stopping-time densities, tree sizes, lower bounds and the distribution of initial iterates, and none decides the problem; its theorems on random models are heuristic support only, whose limits it states (p. 47).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.