RieszWalkSelfSimilarity

Abstract

Finite Riesz moments, orthogonal Laurent polynomials and scale-four projection identities.

Definition 1.1 (prob).

$$\forall (t:\mathbb{N})(S:Set\mathbb{Q}),\mathrm{prob}tS=\sum ^{\prime }(y:\mathbb{N}),\mathrm{if}Nat.cast\left(y\right)/Nat.cast\left(t\right)\in S\mathrm{then}\mathrm{mu}ty\mathrm{else}0$$

Formalization. D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.prob (✓ std3).

Citation. Ryota Hanaoka, Norio Konno (2021). Return probability and self-similarity of the Riesz walk. DOI: 10.26421/QIC21.5-6-5. URL: https://arxiv.org/abs/2010.04518v3.

Commentary.

The sum is over all natural-number sites. Finite propagation makes only finitely many terms nonzero. Rational division gives the normalized site position.

Definition 1.2 (Tset).

$$\forall (nk:\mathbb{N}),\mathrm{Tset}nk=\mathrm{if}k=0\mathrm{then}\{0\}\mathrm{else}Set.Ioc((Nat.cast\left(k\right)-1)/Nat.cast\left(n\right))(Nat.cast\left(k\right)/Nat.cast\left(n\right))$$

Formalization. D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.Tset (✓ std3).

Citation. Ryota Hanaoka, Norio Konno (2021). Return probability and self-similarity of the Riesz walk. DOI: 10.26421/QIC21.5-6-5. URL: https://arxiv.org/abs/2010.04518v3.

Commentary.

Section 4, Conjecture 4.2, arXiv PDF p. 12: “Let ℤ_> = {1, 2, 3, …}. For each n ∈ ℤ_>, we put T₀(n) = {0}, and Tₖ(n) = ((k − 1)/n, k/n] (1 ≤ k ≤ n). Then, for the measure of the Riesz walk with initial state of Ψ₀ = [1, 0]ᵀδ₀, P(X₂ₜ/(2t) ∈ Tₖ(t)) = P(X₈ₜ/(8t) ∈ Tₖ(t)), for any t ∈ ℤ_>.” The case k=0 is a singleton; every other case is left-open and right-closed.

Definition 1.3 (claim).

$$\mathrm{claim}⟺\forall (t:\mathbb{N}),1\le t\to \forall (k:\mathbb{N}),k\le t\to \mathrm{prob}(2\cdot t)(\mathrm{Tset}tk)=\mathrm{prob}(8\cdot t)(\mathrm{Tset}tk)$$

Formalization. D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.claim (✓ std3).

Citation. Ryota Hanaoka, Norio Konno (2021). Return probability and self-similarity of the Riesz walk. DOI: 10.26421/QIC21.5-6-5. URL: https://arxiv.org/abs/2010.04518v3.

Commentary.

Section 4, Conjecture 4.2, arXiv PDF p. 12: “Let ℤ_> = {1, 2, 3, …}. For each n ∈ ℤ_>, we put T₀(n) = {0}, and Tₖ(n) = ((k − 1)/n, k/n] (1 ≤ k ≤ n). Then, for the measure of the Riesz walk with initial state of Ψ₀ = [1, 0]ᵀδ₀, P(X₂ₜ/(2t) ∈ Tₖ(t)) = P(X₈ₜ/(8t) ∈ Tₖ(t)), for any t ∈ ℤ_>.” Here t and k are natural numbers, 1 ≤ t and k ≤ t. The two coordinates at site x are 2x and 2x+1. The intervals have rational endpoints and probabilities are real.

Theorem 1.4 (result).

$$\mathrm{claim}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.result (✓ std3). ∎

Resolves. Problems/hanaoka-konno-2021-riesz-walk-self-similarity (proved) by D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.result.

Source. Repository-derived.

Acknowledgement. Ryota Hanaoka, Norio Konno (2021). Return probability and self-similarity of the Riesz walk. DOI: 10.26421/QIC21.5-6-5. URL: https://arxiv.org/abs/2010.04518v3.

Commentary.

Cumulative probabilities are squared norms of projections onto the Laurent windows. Fourth-power decimation and the parity-weighted projection identity make these norms equal at the two scales. Interval differences give the assertion, including the singleton interval at the origin.

References

• Truth anchor: D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.Tset
• Truth anchor: D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.claim
• Truth anchor: D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.prob
• Truth anchor: D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.result
• Dependency: D5/S3/Quantum/Walks/CMV/Representation
• Dependency: D5/S3/Quantum/Walks/Riesz/Decimation