[Paper Review] Mixing time of fractional random walk on finite fields

Jimmy He, Huy Tuan Pham+1 more|arXiv (Cornell University)|Feb 4, 2021

Algorithms and Data Compression33 references4 citations

TL;DR

This paper studies a non-linear random walk on the finite field 𝔽ₚ defined by Xₙ₊₁ = ι(Xₙ) + εₙ₊₁, where ι(x) = 1/x for x ≠ 0 and ι(0) = 0, and εᵢ are i.i.d. increments. Using comparison with a random walk on P¹(𝔽ₚ) and spectral gap results from SL₂(𝔽ₚ), the authors prove that the mixing time is Θ(log p), demonstrating an exponential speedup over simple random walk on 𝔽ₚ.

ABSTRACT

We study a random walk on $\mathbb{F}_p$ defined by $X_{n+1}=1/X_n+\varepsilon_{n+1}$ if $X_n eq 0$, and $X_{n+1}=\varepsilon_{n+1}$ if $X_n=0$, where $\varepsilon_{n+1}$ are independent and identically distributed. This can be seen as a non-linear analogue of the Chung--Diaconis--Graham process. We show that the mixing time is of order $\log p$, answering a question of Chatterjee and Diaconis.

Motivation & Objective

• To resolve a question posed by Chatterjee and Diaconis regarding the mixing time of a non-linear random walk on 𝔽ₚ defined by Xₙ₊₁ = ι(Xₙ) + εₙ₊₁.
• To establish that this walk mixes in order log p steps, providing an explicit example of exponential speedup via deterministic non-linear updates.
• To connect the dynamics on 𝔽ₚ to a Markov chain on the projective line P¹(𝔽ₚ), which is a quotient of a random walk on SL₂(𝔽ₚ).
• To leverage known spectral gap results for Cayley graphs of SL₂(𝔽ₚ) to derive upper bounds on the mixing time.
• To derive quantitative bounds on the number of solutions to the congruence xy ≡ 1 (mod p) in intervals, using geometric and spectral arguments.

Proposed method

• Use comparison theory from Smith [34] to relate the mixing time on 𝔽ₚ to that on P¹(𝔽ₚ), a quotient space.
• Model the walk on P¹(𝔽ₚ) as a projection of a random walk on the group SL₂(𝔽ₚ) with a fixed generating set.
• Apply the spectral gap result of Bourgain and Gamburd [6] on SL₂(𝔽ₚ) Cayley graphs, which guarantees a constant spectral gap.
• Establish an upper bound on the total variation distance to stationarity using exponential decay from the spectral gap of the projected chain.
• Derive a lower bound via entropy arguments, showing that nH(μ) + log 2 ≤ log p is necessary for mixing, under finite entropy of μ.
• Use Cheeger’s inequality and bottleneck ratio analysis to show that if A ⊆ I and ι(A) ⊆ J for intervals I, J of length m, then |A| ≤ (1−δ)m for some δ > 0, implying limited solution sets to xy ≡ 1 (mod p).

Experimental results

Research questions

• RQ1 What is the order of the mixing time for the fractional random walk Xₙ₊₁ = ι(Xₙ) + εₙ₊₁ on 𝔽ₚ?
• RQ2 Can the exponential speedup in mixing time be rigorously established for this non-linear process compared to simple random walk?
• RQ3 To what extent can spectral gap results on SL₂(𝔽ₚ) be used to bound the mixing time of a projected Markov chain on P¹(𝔽ₚ)?
• RQ4 How many solutions exist to the congruence xy ≡ 1 (mod p) with x ∈ I and y ∈ J for intervals I, J of equal length?
• RQ5 Is it possible to prove that sets A ⊆ I with ι(A) ⊆ J cannot be too large, under mild assumptions on the increment distribution?

Key findings

• The mixing time of the fractional random walk on 𝔽ₚ is Θ(log p), meaning order log p steps are both necessary and sufficient for convergence to uniformity.
• The lower bound on mixing time is established via entropy: ‖Kⁿ(x,·) − π‖_TV ≥ 1 − (nH(μ) + log 2)/log p, valid when H(μ) < ∞.
• The upper bound is proven using spectral gap techniques: for large p, ‖Kⁿ(x,·) − π‖_TV ≤ (√p / 2) e^(-Cn) for some C > 0 depending on μ.
• The proof relies on transferring the problem to a random walk on P¹(𝔽ₚ), which is a quotient of a Cayley graph on SL₂(𝔽ₚ) with a known constant spectral gap.
• The chain Q on 𝔽ₚ defined by Q = PᵀΠPᵀPΠ has a constant spectral gap γ > 0 for large p, implying exponential convergence.
• As a corollary, the number of solutions to xy ≡ 1 (mod p) with x ∈ I and y ∈ J for intervals I, J of length m ≤ p/2 is at most (1−δ)m for some absolute δ > 0 and sufficiently large m, p.