Monotonicity of the elliptic nome $q(t)$ in discrete heat kernel expansions in the quadrant

Elvey Price, Nessmann and Raschel reduced the complete asymptotic expansion of quadrant-walk counts (discrete heat kernels in the quadrant) to theta-function expansions parametrized by the elliptic nome $q$ , which is tied to the counting variable $t$ by an implicit system of theta identities. They proved that $q(t)$ stays in $(0,1)$ , is continuous, and tends to $0$ and $1$ at the two ends of $(0,t_c)$ , where $t_c$ is the critical point of the model, but explicitly left open whether $q(t)$ is increasing, calling the claim only 'plausible' and its proof 'rather technical'. This record states that monotonicity question precisely for the weighted walk model with steps $E,N,S,W,NE$ (weight $a>0$ on the $NE$ step), together with equivalent formulations via the elliptic modulus and the inverse map $t(q)$ , and the problem is open.

Background

Random walks confined to the quarter plane $\mathbb{N}^2$ — lattice paths built from a prescribed finite set of small steps that must never leave the positive quadrant — form a central family of exactly solvable discrete models at the interface of enumerative combinatorics, probability and statistical mechanics. Fix a step set $\mathcal{S}\subset\{-1,0,1\}^2$ with non-negative weights $\omega_s$ , let $q(i,j;n)$ be the total weight of the $n$ -step walks from the origin to $(i,j)$ that stay in the quadrant, and let $Q(x,y;t)=\sum q(i,j;n)x^iy^jt^n$ be the trivariate generating function. After normalization, $q(i,j;n)$ is a discrete analogue of the heat kernel in a cone. By the kernel method, $Q$ satisfies a functional equation whose key object is the kernel curve, the algebraic curve $\mathcal{C}_t=\{(x,y):K(x,y;t)=0\},\qquad K(x,y;t)=xy\bigl(t\,S(x,y)-1\bigr),\qquad S(x,y)=\sum_{s\in\mathcal{S}}\omega_s x^{s_1}y^{s_2},$ which carries the analytic information about the model.

For a non-singular model, $\mathcal{C}_t$ is an elliptic curve (a compact Riemann surface of genus one) for $0<t<t_c$ and degenerates to a rational curve (genus zero) at the critical point $t_c$ , which is simultaneously the smallest positive singularity of $Q(0,0;t)$ and the inverse of the exponential growth rate of the walk counts. Kurkova and Raschel made this picture explicit: the curve admits a uniformization $x=X(z)$ , $y=Y(z)$ by elliptic functions with periods $\pi$ and $\pi\tau(t)$ , $\tau\in i\mathbb{R}_+$ , whose branch points are the zeros $x_1(t),\dots,x_4(t)$ of the discriminant of the kernel viewed as a polynomial in $x$ .

Elvey Price, Nessmann and Raschel reorganized this theory around the Jacobi theta function $\vartheta(z|\tau)$ (an odd entire function with periods $\pi$ and $\pi\tau$ , having the series form $\vartheta(z|\tau)=\sum_{n\ge0}(-1)^ne^{i\pi\tau(n+1/2)^2}(e^{(2n+1)iz}-e^{-(2n+1)iz})$ ) and its nome $q=e^{i\pi\tau}\in(0,1)$ , and used it to compute complete asymptotic expansions of $q(i,j;n)$ as $n\to\infty$ . Their central device is Jacobi's imaginary transformation (the modular involution $\tau\mapsto-1/\tau$ , i.e. $q\mapsto\hat q=e^{-i\pi/\tau}$ ), which converts an ordinary Taylor expansion of the generating function in powers of $q$ , where the series converges rapidly, into an expansion at the critical point $t=t_c$ , where $q\to1$ and $\hat q\to0$ . Applied to the algebraic Kreweras model (the classical step set $\{W,S,NE\}$ , whose generating function is algebraic), the method reproduces the known expansion; applied to the weighted model with step set $\{E,N,S,W,NE\}$ and weight $a>0$ on the $NE$ step — a model whose associated reflection group (the group generated by the branch-swapping involutions of the curve) is infinite, so the orbit-sum techniques available for finite-group models fail, and whose generating function was first treated via Tutte's invariant method by Bernardi, Bousquet-Mélou and Raschel — it yields the first proven occurrence of logarithmic factors $(\log n)^m$ in quadrant-walk asymptotics, with coefficients governed by discrete polyharmonic functions (functions annihilated by a power of the discrete Laplacian of the walk).

The counting variable $t$ enters this machinery only through the nome: for the $NE$ -weighted model the correspondence $t\leftrightarrow q$ is fixed by an implicit system of theta-function identities, and in general $q(t)$ can be written through the complete elliptic integral of the first kind $K(\kappa)=\int_0^{\pi/2}(1-\kappa^2\sin^2\theta)^{-1/2}\,d\theta$ as $q(t)=\exp(-\pi K(\kappa(t))/K(\sqrt{1-\kappa(t)^2}))$ , where the elliptic modulus $\kappa(t)\in(0,1)$ is an explicit cross-ratio of the four branch points $x_1(t),\dots,x_4(t)$ . Throughout, the modulus is this Kurkova–Raschel cross-ratio, under which the nome is strictly decreasing in $\kappa$ ; the classical nome formula in Akhiezer's modulus stated in the source paper's Lemma 4 interchanges the roles of modulus and complementary modulus, and the endpoint limits $q(t)\to0$ as $t\to0$ and $q(t)\to1$ as $t\to t_c$ fix the convention adopted here. What is proven (Lemma 4 of that paper, stated for a class of models broader than the two examples treated) is only that $0<q(t)<1$ on $(0,t_c)$ , that $q$ is continuous there, and that $q(t)\to0$ as $t\to0$ and $q(t)\to1$ as $t\to t_c$ . Immediately after that proof the authors remark that monotonicity 'seems plausible' but is left unproven, noting that a proof would likely come down to the study of the zeros $x_1$ to $x_4$ of the discriminant and would get 'rather technical', and that their main results are deliberately proved without it. Proving monotonicity would promote $q$ to a strictly monotone global reparametrization of $(0,t_c)$ — equivalently, would make $t(q)$ a well-defined increasing function on $(0,1)$ — which is the natural finer analytic control of the parameter mapping underlying the whole theta-function expansion method.

Problem Statement

Prove, or disprove, the following statement. For the weighted quadrant walk model with step set $\{E=(1,0),\,N=(0,1),\,S=(0,-1),\,W=(-1,0),\,NE=(1,1)\}$ , weight $a>0$ on the $NE$ step and weight $1$ on the other four steps (step polynomial $S(x,y)=x+y+x^{-1}+y^{-1}+axy$ ), let $t_c=t_c(a)>0$ be the critical point of the model — the smallest positive singularity of $Q(0,0;t)$ , equivalently the value where the kernel curve $xy(tS(x,y)-1)=0$ drops from genus one to genus zero. The critical point is given by a closed formula in the source paper: writing $a=(1-k^2)/k^3$ with model parameter $k\in(0,1)$ (a decreasing bijection between $k\in(0,1)$ and $a>0$ , so $k$ is uniquely determined by $a$ ), one has $t_c=k/(k^2+3)$ .

Let $q(t)\in(0,1)$ be the elliptic nome of the theta-function uniformization of that curve as defined by Elvey Price, Nessmann and Raschel; equivalently, $q(t)=\exp\left(-\pi\,\frac{K(\kappa(t))}{K\left(\sqrt{1-\kappa(t)^2}\right)}\right),$ where $K$ is the complete elliptic integral of the first kind and $\kappa(t)\in(0,1)$ is the elliptic modulus of the kernel curve, given by the Kurkova–Raschel cross-ratio formula $\kappa(t)=\left(\frac{(x_4(t)-x_1(t))(x_3(t)-x_2(t))}{(x_4(t)-x_2(t))(x_3(t)-x_1(t))}\right)^{1/2}$ in the four zeros $x_1(t),\dots,x_4(t)$ of the discriminant of the kernel polynomial (viewed as a polynomial in $x$ ), numbered as in Kurkova–Raschel. In the theta-function parametrization of the source paper the same correspondence reads: for $0<t<1/(a+4)$ , $q(t)=e^{i\pi\tau(t)}$ , where $(c,\gamma,\tau)$ is the triple with $\gamma,\tau\in i\mathbb{R}$ and $0<\Im(2\gamma)<\Im(\pi\tau)$ satisfying $ac^3=-\frac{\vartheta(4\gamma|\tau)}{\vartheta(2\gamma|\tau)},\qquad c^2=\frac{\vartheta(3\gamma|\tau)}{\vartheta(\gamma|\tau)},\qquad \frac{c}{t}=4\frac{\vartheta'(\gamma|\tau)\,\vartheta(2\gamma|\tau)}{\vartheta'(0|\tau)\,\vartheta(\gamma|\tau)}-2\frac{\vartheta'(2\gamma|\tau)}{\vartheta'(0|\tau)},$ with $\vartheta'$ the derivative of $\vartheta$ in its first argument; the correspondence extends from $(0,1/(a+4))$ to all of $(0,t_c)$ by analytic continuation, as in the source paper (where $\vartheta(z|\tau)$ is the Jacobi theta function defined in the background above).

Claim to settle: for every $a>0$ , the map $t\mapsto q(t)$ is strictly increasing on $(0,t_c)$ . Equivalent formulations of the same fact, any of which may be proved instead: the elliptic modulus $\kappa(t)$ is strictly decreasing on $(0,t_c)$ (the nome is strictly decreasing in the modulus); the inverse correspondence $q\mapsto t(q)$ is well defined and strictly increasing on $(0,1)$ .

A resolution is either (i) a complete proof, covering all $a>0$ and the whole interval $(0,t_c)$ , or (ii) an explicit counterexample: a specific $a>0$ and values $t_1<t_2$ in $(0,t_c)$ with $q(t_1)\ge q(t_2)$ , certified by rigorous numerics. Extending the result to arbitrary non-singular weighted small-step quadrant models in their elliptic regime — the generality in which the authors state their supporting Lemma 4 — is a natural and welcome extension, but the $NE$ -weighted model above is the source-faithful core, and settling it settles the problem.

References

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