Scaling-limit conjecture for longest increasing subsequences

Fix parameters $p_+$ and $p_-$, and let $\tilde\alpha_+$ and $\tilde\alpha_-$ be the critical parameters defined earlier in the paper. Define

$$\mu = m(\tilde\alpha_+;p_+,p_-),\qquad \sigma = \left(\frac{\tilde\alpha_+^2m”(\tilde\alpha_+;p_+,p_-)}{2}\right)^{1/3},$$ $$\mu_+(z)=m(z;p_+,p_-),\qquad \sigma_+(z)=(zm'(z;p_+,p_-))^{1/2},$$ $$\mu_-(z)=m(z;p_-,p_+),\qquad \sigma_-(z)=(zm'(z;p_-,p_+))^{1/2}.$$

Assume that $\tilde\alpha_\pm\notin\{0,\infty\}$, $\sigma>0$, $\sigma_+(\alpha_+)^2>0$ for $\tilde\alpha_+<\alpha_+<R(p_-)^{-1}$, and $\sigma_-(\alpha_-)^2>0$ for $\tilde\alpha_-<\alpha_-<R(p_+)^{-1}$. Write $F_{\mathrm{GUE}}$ for the GUE limiting distribution, $G(w)$ and $H(w_+,w_-)$ for the critical limiting distributions, and $N(a,v)$ for a normal distribution with mean $a$ and variance $v$.

Scaling-limit conjecture. As $n\to\infty$, the following limits hold. If $0\leq\alpha_+<\tilde\alpha_+$ and $0\leq\alpha_-<\tilde\alpha_-$ are fixed, then

$$\frac{\lambda_1(p_+^n,p_-^n;\alpha_+,\alpha_-)-\mu n}{\sigma n^{1/3}}\to F_{\mathrm{GUE}}.$$

Near the critical point, set $\alpha_\pm=\tilde\alpha_\pm\exp(-2w_\pm/(\sigma n^{1/3}))$. If $w_\pm$ and $0\leq\alpha_\mp<\tilde\alpha_\mp$ are fixed, then

$$\frac{\lambda_1(p_+^n,p_-^n;\alpha_+,\alpha_-)-\mu n}{\sigma n^{1/3}}\to G(w_\pm).$$

If $w_+$ and $w_-$ are fixed, then

$$\frac{\lambda_1(p_+^n,p_-^n;\alpha_+,\alpha_-)-\mu n}{\sigma n^{1/3}}\to H(w_+,w_-).$$

Finally, let fixed $\alpha^0_+$ and $\alpha^0_-$ satisfy $\alpha^0_\pm>\tilde\alpha_\pm$ and $\mu_+(\alpha^0_+)=\mu_-(\alpha^0_-)$. Set $\alpha_\pm=\alpha^0_\pm\exp(x_\pm/(\sigma_\pm(\alpha^0_\pm)n^{1/2}))$. If $x_\pm$ and $0\leq\alpha_\mp<\alpha^0_\mp$ are fixed, then

$$\frac{\lambda_1(p_+^n,p_-^n;\alpha_+,\alpha_-)-\mu_\pm(\alpha^0_\pm)n}{n^{1/2}}\to N(x_\pm,\sigma_\pm(\alpha^0_\pm)^2).$$

If $x_+$ and $x_-$ are fixed, then

$$\frac{\lambda_1(p_+^n,p_-^n;\alpha_+,\alpha_-)-\mu_\pm(\alpha^0_\pm)n}{n^{1/2}}\to \max\bigl(N(x_+,\sigma_+(\alpha^0_+)^2),N(x_-,\sigma_-(\alpha^0_-)^2)\bigr).$$

These assertions propose a complete collection of GUE, critical, and Gaussian fluctuation regimes for the longest increasing subsequence statistic under repeated parameters. The critical distributions are compared in the source with distributions studied by Baik and Rains, while the conjecture itself is not stated as proved and its resolution status is therefore open.