A mathematical preprint reports that the global edge-indexed Euler product for an infinitely cusped tree lattice fails coefficient by coefficient. The reason is structural: the model contains infinitely many primitive cycles of length four. The analysis concerns a constructed one-sided-comb quotient, denoted Yq, with q even and at least 2.
That means the global zeta cannot be treated as a formal power series under the model’s edge-indexing rules. The work instead splits the calculation into two pieces: a finite-core determinant for recurrence near a chosen observation set and a height-renormalized finite part for the tail.
The comb exposes the problem
In the one-sided comb, the offending cycles are explicit. For every n ≥ 1, the cycle family Cn has weight q(q − 1). Because infinitely many primitive cycles appear at length four, the coefficient that a global Euler product would assign to that length does not remain finite.
The finite replacement starts with a finite observation set E. The analysis forms a first-return matrix FE(u), which records weighted paths that leave and return to E, and defines ZE(u) = det(IE − FE(u))⁻¹. Its coefficients are finite, so this local zeta is formally well defined even though the global series is not.
At the root, the stationary tail can be reduced algebraically. Eliminating y from the stationary first-passage equations gives a quadratic relation for x, a Schur-complement step that compresses the tail calculation into an algebraic one.
The resulting root local zeta, Za0(u) = 1/(1 − K(u)x(u)), has algebraic degree at most two over Q(q,u). Its exact radius is (q + 1)⁻¹, and its two dominant poles are simple and sit at +(q + 1)⁻¹ and −(q + 1)⁻¹.
Making infinity finite enough to analyze
The infinite tail is handled through the height-damped operator Tρ = BMρ. For 0 < ρ < 1, this operator is trace class, so a Fredholm determinant can be used. The associated expression det(I − uTρ)⁻¹ is meromorphic in u, with an entire reciprocal.
That regulator is paired with an integrated-pressure counterterm and a height-moment series. Across the local disk |u| < (q + 1)⁻¹, both converge normally, and their subtraction has a locally uniform limit. The resulting renormalized finite part is given by convergent product formulas.
The pressure calculation supplies the key separation between the finite core and the tail. Pressure at infinity is defined by deleting finite state cores and taking the pressure of what remains as the core grows. For the two-step multiplicity potential, that tail pressure is log(4q), below the model’s full Gurevich pressure, 2 log(q + 1). The strict gap establishes strong positive recurrence.
That gap is reflected in local orbit growth. The weighted coefficient at n has leading scale (q + 1)^(2n), with an error of order κq^n, where κq is smaller than (q + 1)^2.
What the construction delivers
For sufficiently small |u|, sharp finite-volume pressure limits exist locally uniformly and converge to log Zcell(u). In plain terms, the pressure calculated from finite cutoffs approaches the cell-level expression on that local domain.
The same two-step potential has a unique Ruelle–Perron–Frobenius equilibrium probability measure on the squared root component. The normalized transfer operator also has a spectral-gap realization, giving exponential correlation bounds for the observables covered by the construction.
The renormalized quantity also has a precise height covariance. Replacing h by h + c, with c a nonnegative integer, multiplies the finite part by Zcell(u)⁻ᶜ; a height change supported on finitely many states leaves it unchanged.
The conclusions are tied to Yq and the stated stationary-tail calculation. The local determinants remain local: they retain cycles meeting finite observation sets, not a global zeta, and the renormalized finite part is established on the stated local disk. The document is arXiv:2608.25786v1, dated 26 August 2026, and is supplied as a preprint.
Paper data and sources
Original title: Zeta renormalization and pressure at infinity for an infinitely cusped tree lattice
Authors: Sanghoon Kwon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text