Primitive bridge cancellation
Abstract
Primitive stochastic bridges and the full integer path-difference lattice.
Let V be an arbitrary finite nonempty set, q a natural number, P a real row-stochastic primitive matrix on V, and g an integer q-vector on every ordered vertex pair. A supported edge i to j has P(i,j)>0. Gamma(n,i,j) consists of the actual supported quiver paths of length n from i to j, including the empty path at length zero. Write G(p) for the sum of the edge charges and W(p) for the product of the transition probabilities. The set D contains G(a)-G(b) for every pair of supported paths of equal length and equal endpoints, over all lengths and endpoints. Lambda is the integer span of D.
For a real q-vector x, dot(x,lambda) is the sum of x(k) times lambda(k). The twisted matrix has entries M(x)(i,j)=P(i,j) exp(i dot(x,g(i,j))). The full annihilator U consists of x for which dot(x,lambda) is an integer multiple of 2 pi for every lambda in Lambda. For a finite set F of integer vectors, E(F,x) is the sum over F of 1-cos(dot(x,lambda)). Charges on edges of zero transition probability have no effect on these objects.
Theorem 1.1 (One bridge length and uniform cancellation).
Lean statement: D5/S3/TotalVariation/PrimitiveBridgeCancellation.primitive_bridge_cancellation
Proof. Machine-checked in Lean as D5/S3/TotalVariation/PrimitiveBridgeCancellation.primitive_bridge_cancellation (✓ std3). ∎
Source. Repository-derived.
Commentary.
For any prescribed endpoints iStar and jStar there exist L>0, a finite set F of distinct integer q-vectors, and c>0, all independent of x, such that the integer span of F is Lambda. Every lambda in F is G(a)-G(b) for actual paths a,b in Gamma(L,iStar,jStar), with c<=W(a)W(b). The entry a0=(P^L)(iStar,jStar) is positive. For every real q-vector x, c E(F,x)<=a0^2-|M(x)^L(iStar,jStar)|^2, where the absolute value is the norm of that scalar complex entry. Moreover, E(F,x)=0 if and only if x belongs to the full U. This includes q=0 and the zero lattice, with no gauge or quotient-metric identification.
Fixed-length supported paths are finite because their vertex lists determine them. Splitting off the last supported edge gives the exact matrix-power recurrence, for both P and M(x). Integer finite generation selects finitely many original differences. Primitivity and stochastic row sums provide paths at every sufficiently large length. A common prefix and a common suffix are attached to both members of each selected pair, preserving its difference. Distinct differences make the selected ordered path pairs injective. Their positive weighted cosine deficits can therefore be retained once each from the nonnegative sum over all ordered bridge pairs. Integer span then gives the complete zero set.