On the Albert–Bousquet-Mélou Conjecture for Corners in Quadrant Loops: The Case of Nonnegative Corner Weight

For a real number a let Q_n(a) be the sum of a^c(w) over the walks w of length 2n with steps N, E, S, W that start and end at the origin and stay in the quadrant {x ≥ 0, y ≥ 0}, where c(w) is the number of factors NW and ES of w, and let L_n(a) be the same sum without the quadrant condition. Albert and Bousquet-Mélou conjectured, and Bousquet-Mélou asked in the problem session of Oberwolfach Report 12/2014, that for every a ≥ −1 the series Σ_n Q_n(a)u^n has the same radius of convergence as Σ_n L_n(a)u^n, whose radius they determined. They proved this for a = 1 and a = −1, and for a ≥ 0 they proved that the radius for quadrant loops is at least (2 + √(2+2a))^(−2). We prove the conjecture for every a ≥ 0: the limits of Q_n(a)^(1/n) and of L_n(a)^(1/n) exist and both are equal to (2 + √(2+2a))², and Q_n(a) ≤ (2 + √(2+2a))^(2n) for all n. The proof is elementary and self-contained. It combines a formula for corner-weighted walks under an exponential tilt, a large-deviation bound for the loops that go far to the left of or below the origin, a padding without corners that moves the other loops into the quadrant, and Fekete's lemma; all these tools are standard. The question remains open for −1 < a < 0, and this range contains the values in [−1/3, 0) on which the conjectural growth rate of permutations sortable by two stacks in parallel depends. We show where the argument needs a ≥ 0 and report numerical evidence, not a proof, for the open range. No proof of the case a ≥ 0 was found in the sources we searched; for a ≥ 1 the result seems to follow, after a short argument, from known theorems on weighted walks in cones. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-12872-018 (Bousquet-Mélou, Corners in square lattice loops, Oberwolfach Report 12/2014, Oberwolfach Rep. 11 (2014), p. 711; Albert and Bousquet-Mélou, Europ. J. Combin. 43 (2015), Conjecture 11).