The universal critical-slope conjecture for copolymers near a selective interface

Let $h_c^\mathrm{que}(\beta)$ be the quenched critical curve and let

$$K_c=\lim_{\beta\downarrow 0}\frac{h_c^\mathrm{que}(\beta)}{\beta},$$

when this limit exists, for a copolymer model whose renewal exponent is $\alpha>1$. Critical-slope conjecture. The universal limit satisfies

$$K_c=\frac{1+\alpha}{2\alpha}\quad\text{for all }\alpha>1.$$

The value of $K_c$ is conjectured to be universal, depending only on the renewal exponent $\alpha$ and not on the fine details of the interaction Hamiltonian. Its existence is known for $1<\alpha<2$ under suitable assumptions, while for $\alpha\geq 2$ it remained open in the source; numerical work for simple random walk, corresponding to $\alpha=\tfrac32$, gives $K_c\in[0.82,0.84]$, consistent with the conjectured value $\tfrac56$.