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Local topology, entropy compactness and path-space estimates — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

A fixed label vanishes locally while a spatial mark stays random

Example

For each integer $n \geq 1$ , put $S_{n} = {- n, - n + 1, \dots, n - 1} \subset Z$ and $L_{n} = {1, 2, \dots, 2 n}$ . Let $Ω_{n}$ be the finite set of bijections $π : L_{n} \to S_{n}$ , equipped with its power-set sigma-algebra and probability $P_{n}$ defined by $P_{n} ({π}) = 1 / (2 n)!$ for every $π \in Ω_{n}$ (and by summing these atom masses on every subset); let $π_{n}$ be the identity random bijection on this probability space. The position of fixed label $1$ is $U_{n} = π_{n} (1)$ , and its point measure is $Ξ_{n} = δ_{U_{n}}$ . Give labels $1, \dots, n$ mark $0$ and labels $n + 1, \dots, 2 n$ mark $1$ . Let $V_{n}$ be the mark of the particle occupying spatial site $0$ . For every nonnegative continuous compactly supported $f : R \to R$ and every $ε > 0$ , $P_{n} ⁣ (∣ \int f d Ξ_{n} ∣ > ε) ⟶ 0,$ while $P_{n} (V_{n} = 0) = P_{n} (V_{n} = 1) = 1 / 2$ for every $n$ . The testwise convergence assertion uses the varying-space convention displayed here and requires no coupling between different $n$ .

Example references

Finite-spin product and local topologies

Locally finite counting measures and vague topology

Facts & Assumptions

Given: For each $n$ , the power-set sigma-algebra on $Ω_{n}$ and the uniform probability $P_{n}$ , the finite label set $L_{n}$ , the finite site set $S_{n}$ , and the definitions of finite-spin local convergence and vague convergence.

[F1]

Locally finite counting measures and vague topology: $\int f d δ_{u} = f (u)$ for each nonnegative continuous compactly supported test function; vague convergence is tested by all such functions.

[F2]

Finite-spin product and local topologies: finite-coordinate locality is a separate notion from the spatial vague topology used here.

Verification

1.1givenalgebra

For any fixed site $x \in S_{n}$ , exactly $(2 n - 1)!$ of the $(2 n)!$ bijections send label $1$ to $x$ . Thus $U_{n}$ is uniform on $S_{n}$ under $P_{n}$ . The inverse label $π_{n}^{- 1} (0)$ is also uniform on $L_{n}$ , because exactly $(2 n - 1)!$ bijections assign any prescribed label to site $0$ . Exactly $n$ labels carry each mark, so $P_{n} (V_{n} = 0) = P_{n} (V_{n} = 1) = 1 / 2$ .

2.1givenF1F2step 1.1algebra∎

Let $K = supp f$ . Since $K$ is compact, it is bounded and $K \cap Z$ is finite; write $c_{K} = ∣ K \cap Z ∣$ . Uniformity of $U_{n}$ gives $P_{n} (U_{n} \in K) = ∣ K \cap S_{n} ∣ / (2 n) \leq c_{K} / (2 n)$ . For every $ε > 0$ , [F1] gives $P_{n} ⁣ (∣ \int f d Ξ_{n} ∣ > ε) \leq P_{n} (U_{n} \in K) \leq \frac{c_{K}}{2 n} ⟶ 0.$ Thus each compactly supported vague test tends to zero in probability, while step 1.1 shows that selecting by the fixed spatial site $0$ retains a nondegenerate mark law. This is a statement about spatial point-measure tests; finite-coordinate local convergence on spin configurations is a separate notion by [F2]. These two statements concern different selection rules and do not contradict one another.