Manuscript Oai211-A@1 · Oai211-A.tex · proposition · 3f3e4f63

Proposition: Corner mass and operational transport

Conditional transfer to weighted planar-map blocks › The exact degree-sampling identity · lines 1406–1437 · proof included or linked

Oai211-A@1#prop:corners, Oai211-A@1#eq:degreechannel

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For any nontrivial map in the contract, the pushforward of uniform mass on its $2 n$ nonroot block-tree nodes is

μ_{M} (x) = \frac{de g _{M} ( x )}{2 n} .

If each nonroot node $u$ has a finite channel $Γ_{u}$ , define

Γ_{x} = \frac{1}{de g _{M} ( x )} u \neq = u_{0} π (u) = x \sum Γ_{u} .

Then

x \sum μ_{M} (x) Γ_{x} = \frac{1}{2 n} u \neq = u_{0} \sum Γ_{u} .

Proof. Every nonroot node occupies exactly one parent-block corner, including empty insertions. At each map vertex, the total number of these corners is its number of incident darts. Every dart belongs to exactly one block. A loop contributes two. Thus the pole multiplicity is $de g_{M} (x)$ . Summing the channel mixtures with these multiplicities gives (eq:degreechannel).

Displayed formulas in this passage

Oai211-A@1#eq:degreechannel · line 1421
$x \sum μ_{M} (x) Γ_{x} = \frac{1}{2 n} u \neq = u_{0} \sum Γ_{u} .$
unlabeled · line 1410
$μ_{M} (x) = \frac{de g _{M} ( x )}{2 n} .$
unlabeled · line 1415
$Γ_{x} = \frac{1}{de g _{M} ( x )} u \neq = u_{0} π (u) = x \sum Γ_{u} .$

Proposition: Corner mass and operational transport

Displayed formulas in this passage

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