The Batchelor–Yung critical surface fugacity $y_c(n)=1+2/\sqrt{2-n}$ of the honeycomb half-plane $O(n)$ loop model, and the sharp $T^{-1/4}$ decay of critical bridges

The 2014 paper of Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann proved that the critical surface fugacity for adsorption of self-avoiding walks (the $n=0$ case) on the honeycomb half-plane is $1+\sqrt{2}$, confirming the $n=0$ instance of the 1995 Batchelor–Yung conjecture $y_c(n)=1+2/\sqrt{2-n}$ for the $O(n)$ loop model. Two precisely stated predictions of that paper remain open and form one linked problem: (i) prove (or refute) the Batchelor–Yung conjecture for every loop weight $n\in[-2,2]$ with $n\neq0$, including the corresponding desorbed/adsorbed dichotomy under a justified formulation of the loop-model surface transition; (ii) determine the sharp decay of the height-$T$ bridge generating function $B_T(x_c,1)$ at the self-avoiding-walk critical point, predicted from SLE to be $c\,T^{-1/4}$ (the paper proved only $B_T\to0$). No resolution of either component is known in the literature: the rigorous window for the bridge decay is currently $c/T\le B_T(x_c,1)\le100\,T^{-10^{-10}}$ (Krachun–Panagiotis, Ann. Probab. 2026), and no rigorous surface-transition location is known for any $n\neq0$.

Background

A self-avoiding walk (SAW) is a path on a lattice that never revisits a vertex; it is the standard lattice model of a linear polymer in solution, recovered as the $n\to0$ limit of the $O(n)$ spin model. The honeycomb (hexagonal) lattice is the one two-dimensional lattice where the SAW growth constant (the exponential rate $\mu=\lim_k c_k^{1/k}$ of the number $c_k$ of $k$-step walks) is known exactly: Duminil-Copin and Smirnov (2012) proved Nienhuis' prediction $\mu=\sqrt{2+\sqrt{2}}$ using a parafermionic observable — a generating function over walk configurations weighted by a complex power of the winding angle that satisfies a discrete analogue of the Cauchy–Riemann equations, summed over lattice vertices to give a global identity between generating functions. Smirnov extended the local identity to the $O(n)$ loop model, whose configurations are non-intersecting closed loops on the honeycomb lattice carrying a weight $x$ per occupied vertex and a weight $n\in[-2,2]$ (conventionally $n=2\cos\theta$) per loop; $n=0$ forbids loops and recovers SAWs, $n=1$ is the Ising model in its domain-wall (spin-interface) representation, and $n=-2$ is a Gaussian model. Nienhuis (1982) predicted the bulk critical point $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$, and the $n=0$ case of this is exactly the Duminil-Copin–Smirnov theorem.

Confining the model to a half-plane and giving each occupied boundary (surface) vertex an additional weight, or fugacity, $y$ models polymer adsorption at a wall (physically $y=e^{-\varepsilon/k_BT}$ for a monomer–surface binding energy $\varepsilon$). The free energy per site splits as $f_{\rm bulk}+f_{\rm surface}+\cdots$, and a surface phase transition occurs at a critical value $y_c$: for $y<y_c$ the polymer is desorbed (zero density in the surface), for $y>y_c$ it is adsorbed (positive surface density). At the bulk critical point $x_c$ this is an instance of the special surface transition of surface critical phenomena. For SAWs the critical fugacity is defined analytically following Hammersley, Torrie and Whittington (1982) via $\mu(y)=\lim_k(c_k^+(y))^{1/k}$, where $c_k^+(y)$ is the partition function of $k$-vertex half-plane walks with weight $y$ per surface contact; $y_c$ is the threshold beyond which $\mu(y)>\mu$. In 1995 Batchelor and Yung, using the integrability of the boundary loop model (reflection matrices) together with numerics, conjectured that for the honeycomb $O(n)$ loop model at its bulk critical point the transition occurs at $y_c(n)=1+2/\sqrt{2-n}$ for all $n\in[-2,2]$.

The source paper (Beaton, Bousquet-Mélou, de Gier, Duminil-Copin, Guttmann, Comm. Math. Phys. 326 (2014) 727–754) proved the $n=0$ case, $y_c(0)=1+\sqrt{2}$. It first generalized the parafermionic local identity to a half-plane carrying a surface fugacity, valid for all $n\in[-2,2]$, and derived a global identity between boundary generating functions in which the value $y^*(n)=1+2/\sqrt{2-n}$ plays the same distinguished role that the growth constant plays in Smirnov's identity — evidence for, but not a proof of, the conjecture when $n\neq0$. For $n=0$ the proof combines the global identity with a strip analysis: walks confined to a strip of height $T$ are classified as arches (ending on the starting side) or bridges (ending on the far side); the strip generating functions $A_T(x,y)$ and $B_T(x,y)$ have a radius of convergence $y_T$ in $y$ that decreases to $y_c$ as $T\to\infty$. The third ingredient, of independent interest, is that the bridge generating function at criticality, $B_T(x_c,1)$, tends to $0$ (Theorem 10 of the paper), proved probabilistically by decomposing bridges at renewal points into irreducible bridges (the classical renewal structure of bridges) and a geometric 'stickbreak' construction, building on sub-ballisticity of self-avoiding walk (Duminil-Copin–Hammond 2013), i.e. the fact that the endpoint displacement of a uniformly random $n$-step self-avoiding walk is $o(n)$. Two predictions, however, were explicitly left open and have stayed open: the full conjecture for loop weight $n\neq0$, and the decay rate of $B_T(x_c,1)$ — predicted from the Lawler–Schramm–Werner analysis of the conjectural SLE$_{8/3}$ scaling limit of planar SAW (SLE, the Schramm–Loewner Evolution, is the family of random curves describing scaling limits of critical two-dimensional lattice models) to be $T^{-1/4}$, while only $B_T(x_c,1)\to0$ is proved. The paper also notes that for other lattices (square, triangular) no conjectured values of the critical surface fugacity exist, and that the honeycomb lattice with rotated boundary was settled separately by Beaton (2014), proving the 1998 conjecture of Batchelor, Bennett-Wood and Owczarek.

Problem Statement

The problem has two components, both drawn from the unresolved content of the source paper; they jointly define one objective — the rigorous surface-adsorption picture of the half-plane honeycomb $O(n)$ loop model.

Setup. Let $\mathbb{H}$ be the honeycomb lattice cut into a half-plane by a horizontal line, whose boundary vertices are surface vertices. For $n\in[-2,2]$ (conventionally $n=2\cos\theta$, $\theta\in[0,\pi]$), the $O(n)$ loop model on $\mathbb{H}$ assigns to a configuration $\gamma$ — a collection of non-intersecting closed loops, together with one self-avoiding walk component (defect) whose endpoints lie on the boundary — the weight $x^{|\gamma|}n^{\ell(\gamma)}y^{c(\gamma)}$, where $|\gamma|$ is the number of occupied vertices, $\ell(\gamma)$ the number of loops, and $c(\gamma)$ the number of occupied surface vertices. Fix the bulk activity at Nienhuis' critical value $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$, so that $y$ is the only free parameter.

Part (i) — the Batchelor–Yung conjecture for $n\neq0$. Prove, or rigorously refute, the statement of Conjecture 1 of the source paper: the model undergoes its surface transition at

$y_c(n)=1+\frac{2}{\sqrt{2-n}}.$

For every $n\in[-2,2]$ with $n\neq0$, define the loop-model surface transition precisely, justify that definition, and determine the transition point, proving the full desorbed ($y<y_c(n)$) / adsorbed ($y>y_c(n)$) dichotomy with the transition exactly at $y_c(n)=1+2/\sqrt{2-n}$ (the degenerate endpoint $n=2$, where the conjectured value diverges, may be treated in the natural limiting sense). The case $n=0$, $y_c=1+\sqrt{2}$, is already proved and is excluded. A result covering only a strict subset of loop weights does not determine the surface transition for all $n\in[-2,2]$ with $n\neq0$.

Part (ii) — sharp decay of critical bridges at $n=0$. For the $n=0$ model (self-avoiding walks), let $B_T(x,1)$ be the generating function of bridges of height $T$: self-avoiding walks confined to a strip of height $T$ that start at a mid-edge (the midpoint of a lattice edge, the standard convention for walk endpoints on the honeycomb lattice) immediately below the bottom boundary and end on the top boundary, weighted $x$ per occupied vertex, with no surface fugacity. Determine the asymptotic behaviour of $B_T(x_c,1)$ as $T\to\infty$ at $x_c=1/\sqrt{2+\sqrt{2}}$. The prediction from the conjectural $\mathrm{SLE}_{8/3}$ scaling limit of planar self-avoiding walk is

$B_T(x_c,1)\;\sim\;c\,T^{-1/4}$

with $0<c<\infty$. Prove the predicted behaviour, or rigorously establish the true asymptotics when they differ from the prediction. The $\mathrm{SLE}_{8/3}$ scaling limit itself may not be assumed unless independently proved. For reference, the currently known rigorous window is $c/T\le B_T(x_c,1)\le100\,T^{-10^{-10}}$.

Parts (i) and (ii) are the two unresolved predictions of the source paper about the same surface transition, and they are jointly posed: each admits either a positive resolution (a proof) or a negative one (a rigorous refutation that determines the true behaviour).

References

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