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 \(\{x\ge0,\ y\ge0\}\), 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 \(a\ge-1\) the series \(\sum_nQ_n(a)u^n\) has the same radius of convergence as \(\sum_nL_n(a)u^n\), whose radius they determined. They proved this for \(a=1\) and \(a=-1\), and for \(a\ge0\) they proved that the radius for quadrant loops is at least \((2+\sqrt{2+2a})^{-2}\). We prove the conjecture for every \(a\ge0\): the limits of \(Q_n(a)^{1/n}\) and of \(L_n(a)^{1/n}\) exist and both are equal to \((2+\sqrt{2+2a})^2\), and \(Q_n(a)\)\({}\le(2\)\({}+\sqrt{2+2a})^{2n}\) 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 \(a\ge0\) and report numerical evidence, not a proof, for the open range. No proof of the case \(a\ge0\) was found in the sources we searched; for \(a\ge1\) the result seems to follow, after a short argument, from known theorems on weighted walks in cones. This is an unrefereed note.