DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Simple symmetric walk on the integer lattice

Definition

For an integer $d \geq 1$ , the simple symmetric nearest-neighbor transition matrix on $Z^{d}$ is

$p (z, z + e_{j}) = p (z, z - e_{j}) = \frac{1}{2 d} (j = 1, \dots, d),$

with every other entry zero. Here $e_{j}$ is the $j$ th standard basis vector. The $2 d$ neighbors $z \pm e_{j}$ are distinct when $d \geq 1$ , so each row sums to $2 d / (2 d) = 1$ ; this is a transition matrix in the sense of Transition matrices and n-step probabilities. The definition uses no Choice.

Depends on

Transition matrices and n-step probabilities

Used by

Higher-dimensional simple symmetric walks are transient Corollary
One-dimensional simple symmetric walk is recurrent Corollary
Two-dimensional simple symmetric walk is recurrent Corollary

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
Levin, Peres and Wilmer, Markov Chains and Mixing Times, second edition (standard reference, not scraped)