Multicritical Schur Measures and Higher-Order Analogues of the Tracy–Widom Distribution

Abstract

We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form $1/(2m+1)$, with m a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy–Widom GUE distribution recovered for $m=1$. We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz.

Appendices

A. Reminders on Schur Measures

In this appendix we recall the following seminal result:

Theorem 15

(Determinantal point process associated with the Schur measure [10]) Fix two sequences $t= (t_1,t_2,\ldots )$ and $t'= (t'_1,t'_2,\ldots )$ such that $\mathbb {P}(\lambda ) := e^{-\sum _r rt_r t_r'} s_\lambda [t]s_\lambda [t']$ is a Schur measure, and let ${\lambda }$ be a random partition under that measure. Then, for each finite set $\{k_1,\ldots ,k_n\} \subset \mathbb {Z}+\tfrac{1}{2}$, we have

$$\begin{aligned} \mathbb {P}(\{k_1,\ldots ,k_n\} \subset S({\lambda })) =\rho _n(k_1,\ldots , k_n) = \det _{1 \le i,j \le n} K(k_i,k_j) \end{aligned}$$

(A.1)

where

$$\begin{aligned} K(k,\ell ) = \sum _{i=0}^\infty J_{k+i+1/2}(t,t') J_{\ell +i+1/2}(t,t') \end{aligned}$$

(A.2)

where $J_n(t,t')$ is the multivariate Bessel function

$$\begin{aligned} J_n(t,t') = \frac{1}{2\pi i} \oint \exp \bigg [\sum _r t_r z^r-\sum _rt'_r z^{-r}\bigg ] \frac{dz}{z^{n+1}}. \end{aligned}$$

(A.3)

The kernel K is generated by

$$\begin{aligned} \sum _{k,\ell \in \mathbb {Z}+\frac{1}{2}} z^{k}w^{-\ell } K(k,\ell ) = \frac{\exp \big [\sum _r t_r z^r-\sum _rt'_r z^{-r}\big ]}{\exp \big [\sum _r t_r w^r - \sum _rt'_r w^{-r} \big ]}\frac{\sqrt{zw}}{z-w},\quad |w|<|z|. \end{aligned}$$

(A.4)

This is summarised in the Hermitian case $t' = t^*$ in Sect. 2.1, by way of a lattice fermion model. Here we use the same anti-commuting operators and partition-indexed vectors to define a determinantal point process, but in a self-contained way without reference to quantum mechanics. Following [10, Appendix A], we consider the space spanned by the vectors $|\mathbin {S} \rangle$ indexed by sets of distinct half-integers $S\subset \mathbb {Z}+\frac{1}{2}$ (making a change of notation from Sect. 2.1) such that both the set $S {\setminus } (\mathbb {Z}_{\le 0} - \frac{1}{2})$ of positive half-integers in S and the set $(\mathbb {Z}_{\le 0} - \frac{1}{2})\setminus S$ of negative half-integers not in S are finite. We equip this space with the inner product

$$\begin{aligned} \langle \mathbin {S}|\mathbin {T} \rangle = \delta _{S,T}. \end{aligned}$$

(A.5)

We define the action of the creation and annihilation operators $c_k^\dagger , c_k$ on the vectors $|\mathbin {S} \rangle$ by

$$\begin{aligned} \ c_k^\dagger |\mathbin {S} \rangle = {\left\{ \begin{array}{ll} (-1)^{N_k}|\mathbin {S \cup \{k\}} \rangle &{}\text {if }k \notin S \\ 0 &{} \text {if }k \in S \end{array}\right. },\, c_k|\mathbin {S} \rangle = {\left\{ \begin{array}{ll} 0 &{}\text {if }k \notin S \\ (-1)^{N_k}|\mathbin {S \setminus \{k\}} \rangle &{}\text {if }k \in S \end{array}\right. } \end{aligned}$$

(A.6)

where $N_k:=|S\setminus (\mathbb {Z}_{<k}+\frac{1}{2} )|$ is the number of elements greater than k in S. Hence, $c_k^\dagger$ and $c_k$ are adjoint with respect to $\langle \mathbin {\cdot }|\mathbin {\cdot } \rangle$, and the orthonormalisation of the basis $\{|\mathbin {S} \rangle \}$ ensures that they must satisfy the canonical anti-commutation relations (2.1).

Fig. 8

The Young diagram of the partition $\lambda = (4,3,1)$, with the corresponding fermion configuration $S(\lambda ) = (\frac{7}{2},\frac{3}{2},-\frac{3}{2},-\frac{7}{2},-\frac{9}{2},-\frac{11}{2}, \ldots )$ shown below. The darker boxes form a ribbon of length 4, and adding this ribbon to $\mu = (2,1,1)$ corresponds to moving the fermion at position $-\frac{1}{2}$ in $S(\mu ) =(\frac{3}{2},-\frac{1}{2},-\frac{3}{2},-\frac{7}{2},-\frac{9}{2},-\frac{11}{2}, \ldots )$ to position $\frac{7}{2}$

In terms of the set $S(\lambda )$ defined at (3.1), the partition-indexed vectors already defined at (2.9) are $|\mathbin {\lambda } \rangle := |\mathbin {S(\lambda )} \rangle$, and in particular the vector corresponding to the empty partition (or domain wall state) is $|\mathbin {\emptyset } \rangle := |\mathbin {S(\emptyset )} \rangle$ is indexed by the negative half-integers $S(\emptyset ) = \{-\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{5}{2},\ldots \}$. For all $\lambda$ we have

$$\begin{aligned} |S(\lambda ) \setminus (\mathbb {Z}_{\le 0} - \tfrac{1}{2})| = | (\mathbb {Z}_{\le 0} - \tfrac{1}{2}) \setminus S(\lambda )|. \end{aligned}$$

(A.7)

The bosonic creation and annihilation operators $a_{\pm r}$ defined at (2.4) preserve $|S {\setminus } (\mathbb {Z}_{\le 0} - \tfrac{1}{2})| - | (\mathbb {Z}_{\le 0} - \tfrac{1}{2}) {\setminus } S|$ when acting on a state $|\mathbin {S} \rangle$, and their action on the state $|\mathbin {\lambda } \rangle$ has a natural Young-diagrammatic interpretation: we have

$$\begin{aligned} a_{-r} |\mathbin {\lambda } \rangle = \sum _{\mu = \lambda + \square ^r } |\mathbin {\mu } \rangle \end{aligned}$$

(A.8)

where the sum is taken over all partitions $\mu$ whose Young diagrams differ from that of $\lambda$ by the addition of a “ribbon” of length r, and in particular $(a_1)^n |\mathbin {\emptyset } \rangle = \sum _{\lambda : |\lambda | = n} |\mathbin {\lambda } \rangle$; see Fig. 8.

Proof of Theorem 15

We first write the Schur measure in terms of inner products on the vector space described above. Fix two sequences $t = (t_1, t_2, \ldots )$ and $t' = (t_1',t_2',\ldots )$, and let

$$\begin{aligned} \Gamma _{\pm }(t):= \exp \bigg [ \sum _{r\ge 1} t_{r} a_{{\pm }r}\bigg ]. \end{aligned}$$

(A.9)

Note that $\Gamma _+(t) |\mathbin {\emptyset } \rangle = |\mathbin {\emptyset } \rangle$ and $\langle \mathbin {\emptyset }|\Gamma _-(t) = \langle \mathbin {\emptyset }|$.

Lemma 16

For any partition $\lambda$, we have

$$\begin{aligned} \langle \mathbin {\emptyset }| \Gamma _+(t) |\mathbin {\lambda } \rangle = s_\lambda [t] \end{aligned}$$

(A.10)

and the Schur measure may be written

$$\begin{aligned} \mathbb {P}(\lambda ) = \frac{1}{Z} \langle \mathbin {\emptyset }| \Gamma _+(t) |\mathbin {\lambda } \rangle \langle \mathbin {\lambda }| \Gamma _-(t') |\mathbin {\emptyset } \rangle \end{aligned}$$

(A.11)

where the normalisation is $Z = \langle \mathbin {\emptyset }| \Gamma _+(t) \Gamma _-(t') |\mathbin {\emptyset } \rangle = e^{\sum _r r t_r t_r'}$.

Proof

From the anti-commutation relations (2.1), we have

$$\begin{aligned}{}[a_r,c^\dagger (z)] = z^rc^\dagger (z),\qquad [a_r,c(w)] = -cw^r c(w), \end{aligned}$$

(A.12)

in terms of the generating functions

$$\begin{aligned} c^\dagger (z):= \sum _k z^k c^\dagger _k \quad \text {and} \quad c(w):= \sum _\ell w^{-\ell } c_\ell . \end{aligned}$$

(A.13)

Then, from the formula

$$\begin{aligned} e^{A}B = \sum _{n=0}^\infty \frac{1}{n!} \underbrace{[A,[A,\ldots [A}_{n\text { times}},B]\ldots ]] e^A \end{aligned}$$

(A.14)

we obtain

$$\begin{aligned} \Gamma _{\pm } (t)c^\dagger (z)= & {} e^{\sum _r t_r z^{\pm r}} c^\dagger (z) \Gamma _{\pm }(t),\nonumber \\ \Gamma _{\pm }(t) c(w)= & {} e^{-\sum _r t_r z^{\pm r}} c(w) \Gamma _{\pm }(t). \end{aligned}$$

(A.15)

Recalling that $e^{\sum _r t_r z^r} =:\sum _i h_i[t] z^i$ generates the complete homogeneous symmetric functions as defined in (1.2), we extract coefficients to recover

$$\begin{aligned} \Gamma _+(t) c^\dagger _k&= \sum _{i=0}^\infty h_i[t]c^\dagger _{k-i} \Gamma _+(t) =: {\hat{c}}^\dagger _k \Gamma _+(t), \nonumber \\ \Gamma _+(t) c_k&= \sum _{m=0}^\infty h_i[t]c_{k+i} \Gamma _+(t) =: {\hat{c}}_k \Gamma _+(t), \end{aligned}$$

(A.16)

so ${\hat{c}}_k^\dagger$ and ${\hat{c}}_k$ are linear combinations of the $c^\dagger _k$ and $c_k$ respectively. Hence, we can apply Wick’s lemma [53] to obtain

$$\begin{aligned} \langle \mathbin {\emptyset }| \Gamma _+(t) |\mathbin {\lambda } \rangle&= \langle \Gamma _+(t) c^\dagger _{\lambda _1 - \frac{1}{2}} c_{- \frac{1}{2}} c^\dagger _{\lambda _2 - \frac{3}{2}} c_{-\frac{3}{2}} \ldots c^\dagger _{\lambda _{\ell (\lambda )} - \ell (\lambda ) + \frac{1}{2}} c_{-\ell (\lambda ) + \frac{1}{2}} \rangle \nonumber \\&= \langle {\hat{c}}^\dagger _{\lambda _1 - \frac{1}{2}} {\hat{c}}_{- \frac{1}{2}} {\hat{c}}^\dagger _{\lambda _2 - \frac{3}{2}} {\hat{c}}_{-\frac{3}{2}} \ldots {\hat{c}}^\dagger _{\lambda _{\ell (\lambda )} - \ell (\lambda ) + \frac{1}{2}} {\hat{c}}_{-\ell (\lambda ) + \frac{1}{2}} \rangle \nonumber \\&= \det _{1\le i,j \le \ell (\lambda )} \langle {\hat{c}}^\dagger _{\lambda _i - i + \frac{1}{2}} {\hat{c}}_{-j + \frac{1}{2}} \rangle \end{aligned}$$

(A.17)

Since the complete homogeneous functions satisfy $\sum _i h_{n-i}[t] h_{i}[t] = h_n[t]$, the matrix element is

$$\begin{aligned} \langle {\hat{c}}^\dagger _{\lambda _i - i + \frac{1}{2}} {\hat{c}}_{-j + \frac{1}{2}} \rangle&= \sum _{m, n} h_m[t] h_n[t] \delta _{\lambda _i - i -m,n-j } \nonumber \\&= \sum _{n} h_{\lambda _i - i + j - n}[t] h_n[t] = h_{\lambda _i - i + j}[t]; \end{aligned}$$

(A.18)

recalling the expression (1.1) for the Schur function, we have

$$\begin{aligned} \langle \mathbin {\emptyset }| \Gamma _+(t) |\mathbin {\lambda } \rangle = \det _{1 \le i,j \le \ell (\lambda )} h_{\lambda _i - i + j} = s_\lambda [t] \end{aligned}$$

(A.19)

as required. Since we similarly have $\langle \mathbin {\lambda }| \Gamma _-(t') |\mathbin {\emptyset } \rangle$, we have

$$\begin{aligned} \sum _{\lambda } s_\lambda [t] s_\lambda [t'] = \sum _{\lambda }\langle \mathbin {\emptyset }| \Gamma _+(t) |\mathbin {\lambda } \rangle \langle \mathbin {\lambda }| \Gamma _-(t') |\mathbin {\emptyset } \rangle = \langle \mathbin {\emptyset }| \Gamma _+(t) \Gamma _-(t') |\mathbin {\emptyset } \rangle \end{aligned}$$

(A.20)

as the sum of projections $\sum _\lambda |\mathbin {\lambda } \rangle \langle \mathbin {\lambda }|$ is simply the identity.

By application of the Baker–Campbell–Hausdorff formula $e^A e^B = e^{[A,B]}e^Be^A$ where $[A,[A,B]] = [B,[A,B]]=0$, we have

$$\begin{aligned} \Gamma _+(t)\Gamma _-(t') = e^{\sum _r r t_rt_r'} \Gamma _-(t')\Gamma _+(t) \end{aligned}$$

(A.21)

and hence the normalisation is $Z = \langle \mathbin {\emptyset }| \Gamma _+(t) \Gamma _-(t') |\mathbin {\emptyset } \rangle = e^{\sum _r r t_rt_r'}$, giving the expression for the Schur measure required.

Now, consider the random set of distinct half integers $S(\lambda )$ where $\lambda$ is distributed by the Schur measure. From the expression (A.11) for the Schur measure, the n-point correlation function on this set is⁶

$$\begin{aligned} \mathbb {P}(\{k_1,\ldots ,k_n\} \subseteq S(\lambda ) )= & {} \rho _n(k_1,\ldots ,k_n) \nonumber \\= & {} \frac{1}{Z}\langle \mathbin {\emptyset }| \Gamma _+(t) c_{k_1}^\dagger c_{k_1} \cdots c_{k_n}^\dagger c_{k_n} \Gamma _-(t')|\mathbin {\emptyset } \rangle \end{aligned}$$

(A.22)

for any finite set of half-integers $\{k_1,\ldots ,k_n\}$. We will use the notation $\langle \cdot \rangle := \langle \mathbin {\emptyset }| \cdot |\mathbin {\emptyset } \rangle$ for the expectation on the domain wall state.

Lemma 17

We have

$$\begin{aligned} \rho _n(k_1,\ldots ,k_n) = \det _{1 \le i,j \le n} K(k_i,k_j) \end{aligned}$$

(A.23)

for a kernel

$$\begin{aligned} K(k,\ell ) = \langle \mathbin {\emptyset }| \Gamma _+(t) \Gamma _-(t')^{-1} c_k^\dagger c_\ell \Gamma _-(t')\Gamma _+(t)^{-1}|\mathbin {\emptyset } \rangle \end{aligned}$$

(A.24)

which is given by (A.2), and has generating function (A.4).

Proof

Setting

$$\begin{aligned} {\tilde{c}}_{k}^\dagger = \Gamma _+(t) \Gamma _-(t')^{-1}c^\dagger _k \Gamma _-(t')\Gamma _+(t)^{-1}, \quad {\tilde{c}}_{k} = \Gamma _+(t) \Gamma _-(t')^{-1}c_k \Gamma _-(t')\Gamma _+(t)^{-1},\nonumber \\ \end{aligned}$$

(A.25)

we have

$$\begin{aligned} \rho _n(k_1,\ldots ,k_n)&= \frac{1}{Z}\langle \Gamma _+(t) \Gamma _-(t')\Gamma _+(t)^{-1} {\tilde{c}}_{k_1}^\dagger {\tilde{c}}_{k_1} \cdots {\tilde{c}}_{k_n}^\dagger {\tilde{c}}_{k_n} \Gamma _+(t) \Gamma _-(t')^{-1} \Gamma _-(t')\rangle \nonumber \\&= \langle {\tilde{c}}_{k_1}^\dagger {\tilde{c}}_{k_1} \cdots {\tilde{c}}_{k_n}^\dagger {\tilde{c}}_{k_n} \rangle . \end{aligned}$$

(A.26)

Note that by (A.15), the ${\tilde{c}}_k$ are linear combinations of the $c_k$. We can therefore apply Wick’s lemma to obtain

$$\begin{aligned} \rho _n(k_1,\ldots ,k_n) = \det _{1\le i,j\le n} \langle {\tilde{c}}_{k_i}^\dagger {\tilde{c}}_{k_j} \rangle . \end{aligned}$$

(A.27)

The generating function of $K(k,\ell )$ is, from (A.15),

$$\begin{aligned} \sum _{k,\ell } z^k w^{-\ell } K(k,\ell )&= \langle \Gamma _+(t) \Gamma _ -(t')^{-1}c^\dagger (z) c(w) \Gamma _-(t')\Gamma _+(t)^{-1}\rangle \nonumber \\&= e^{\sum _r t_r z^{r}- t_r z^{-r}}\langle c^\dagger (z) c(w)\rangle e^{\sum _r t_r' w^{-r}-\sum _r t_r w^r}. \end{aligned}$$

(A.28)

To obtain an explicit expression, we evaluate the term $\langle c^\dagger (z) c(w) \rangle$ and get, for $|w|<|z|$,

$$\begin{aligned} \sum _{k,\ell \in \mathbb {Z}+\frac{1}{2}} \frac{z^k}{w^\ell } \langle c^\dagger _k c_\ell \rangle = \sum _{k < 0 } \frac{z^k}{w^\ell } \delta _{k=\ell } = \frac{\sqrt{zw}}{z-w}, \end{aligned}$$

(A.29)

which gives (A.4) as required. To write $K(k,\ell )$ in terms of the multivariate Bessel functions defined at (A.3), we manipulate the formal series in (A.28) further and get

$$\begin{aligned} \sum _{k,\ell } z^k w^{-\ell } K(k,\ell )&= \sum _{k,\ell }z^k w^{-\ell }\sum _{m\in \mathbb {Z}} z^{m} J_{m}(t,t') \sum _{n\in \mathbb {Z}} w^{-n} J_{n}(t,t') \varvec{1}_{k=\ell , k<0} \nonumber \\&= \sum _{i=0 }^\infty \sum _{m,n \in \mathbb {Z}} z^{m-i-\frac{1}{2}} J_{m}(t,t') w^{i-n+ \frac{1}{2}} J_{n}(t,t') \nonumber \\&= \sum _{k,\ell } z^{k}w^{-\ell } \sum _{i=0}^\infty J_{k+i+\frac{1}{2}}(t,t') J_{\ell +i+\frac{1}{2}}(t,t') . \end{aligned}$$

(A.30)

This recovers (A.2) as required.

B. Cylindric Multicritical Schur Measures and Positive Temperature Edge Fluctuations

In [6], the authors found a direct generalisation of the higher-order TW-GUE distribution for the fluctuations in the largest momentum in a grand canonical ensemble of fermions in a 1D flat trap potential at positive temperature. Here, we will construct a discrete model with the same asymptotic edge behaviour, as an instance of the periodic Schur process [54]. Indeed, it was shown in [40] that the periodic Schur process can be interpreted as a system of fermions at positive temperature (the discussion in Sect. 2.1 corresponding to the zero temperature case). In particular, the positive temperature generalization of the Poissonised Plancherel measure is a measure on pairs of partitions which gives rise to fluctuations governed by Johansson’s positive temperature generalisation of the TW-GUE distribution [22] in a suitable asymptotic regime (see [40, Theorem 1.1]).

We may similarly generalise the multicritical Schur measures to the positive temperature setting. Let $\lambda$ and $\mu$ be two partitions and let $t = (t_1,t_2,\ldots )$ be sequence of Miwa times. The skew Schur function $s_{\lambda /\mu }[t]$ is defined via the Jacobi–Trudi identity as

$$\begin{aligned} s_{\lambda /\mu }[t] = \det _{1 \le i,j \le \ell (\lambda )} h_{\lambda _i - i -\mu _j + j}[t] \end{aligned}$$

(B.1)

where $\sum _k h_k[t]z^k = \exp [ \sum _{r\ge 1} t_r z^r]$ as in (1.2). Note that $s_{\lambda /\mu } = 0$ if $\lambda _i < \mu _i$ for some i. Then, we have the following definition:

Definition 18

(Cylindric multicritical measure) Let $\gamma = (\gamma _1,\gamma _2,\ldots )$ be a sequence of real numbers defining an order m multicritical measure by the conditions of Definition 0 with right edge and fluctuation coefficients b, d, let $\theta$ and u be non-negative parameters with $u<1$. Then, the measure on pairs of partitions $(\lambda ,\mu )$

$$\begin{aligned} \mathbb {P}^{m}_{u,\theta } (\lambda ,\mu ) = \frac{1}{Z}u^{|\mu |}s_{\lambda /\mu }[\theta \gamma ]^2, \qquad Z = \frac{\exp [\frac{\theta ^2}{1-u} \sum _{r} r^2 \gamma _r^2]}{\prod _{i\ge 1}(1-u^i)} \end{aligned}$$

(B.2)

is called an order m cylindric multicritical measure.

From the partition function Z, we see that

$$\begin{aligned} {\mathbb {E}}^{m}_{u,\theta }(|{\lambda }|) = \frac{\theta ^2}{(1-u)^2} \sum _r r^2 \gamma _r^2 - u \frac{d}{du}\log (u;u)_{\infty }. \end{aligned}$$

(B.3)

Since $\log (u;u)_{\infty } \sim -\frac{\pi ^2}{6(1-u)}$ as $u \rightarrow 1$, we see that the first term dominates for $\theta \rightarrow \infty$, whether u is fixed or tends to 1. Hence, $\Theta := \theta /(1-u)$ asymptotically defines a natural length scale for the parts ${\lambda }_i,{\lambda }_i'$.

For the cylindric multicritical measures, we have the following positive temperature generalisation of Theorem 1, which is also a multicritical generalisation of [40, Theorem 1.1]:

Theorem 19

(Asymptotic edge fluctuations of cylindric multicritical measures) Let $(\lambda ,\mu )$ be a random pair of partitions under a cylindric multicritical measure $P^{m}_{u,\theta }$ with right edge and fluctuation coefficients b, d. Then, for any $\alpha >0$, in the critical scaling regime $\theta \rightarrow \infty$, $u \rightarrow 1$ with $\theta (1-u)^{2m} \rightarrow \alpha ^{2m+1}d >0$, we have

$$\begin{aligned} \mathbb {P}_{u,\theta }^{m} \left( \frac{{\lambda }_1 - b \Theta }{(d \Theta )^{\frac{1}{2m+1}}} < s\right) \rightarrow F_{2m+1}^\alpha (s):= \det (1-\mathcal {A}_{2m+1}^\alpha )_{L^2([s,\infty ))} \end{aligned}$$

(B.4)

with $\Theta :=\frac{\theta }{1-u} \sim \big (\frac{\theta }{\alpha }\big )^{\frac{2\,m+1}{2\,m}} d^{-\frac{1}{2\,m}}$ and $F_{2\,m+1}^\alpha$ the Fredholm determinant of the higher-order $\alpha$-Airy integral kernel

$$\begin{aligned} \mathcal {A}_{2m+1}^\alpha (x,y):= \int _{-\infty }^\infty \frac{e^{\alpha v}}{1+ e^{\alpha v}} {{\,\textrm{Ai}\,}}_{2m+1} (x+ v) {{\,\textrm{Ai}\,}}_{2m+1}(y+v) dv. \end{aligned}$$

(B.5)

Here again, $\alpha$ plays the role of a limiting inverse temperature, and in the limit $\alpha \rightarrow \infty$ we have $F_{2m+1}^\alpha \rightarrow F_{2m+1}$. We note that critical exponents are unchanged by the passage to finite temperature in this regime once we replace the large parameter $\theta$ with $\Theta$, which also tends to infinity. The Fredholm determinants $F_{2m+1}^\alpha$ have been related to an integro-differential generalisation of the Painlevé II hierarchy by Krajenbrink [55], who generalised an approach of Amir, Corwin and Quastel [56] from the $m=1$ case, and by Bothner, Cafasso and Tarricone [57], who used a rigorous Riemann–Hilbert approach.

Determinantal point process in the grand canonical ensemble Periodic Schur processes are in general not determinantal, as first observed by Borodin [54], who showed how to remedy to this issue via a procedure called shift-mixing. In the language of fermions, this amounts to passing to the grand canonical ensemble [40]. Applying this procedure to the cylindric multicritical measure $\mathbb {P}_{u,\theta }^{m}$, we find that the shifted half-integer set

$$\begin{aligned} S_{{c}}({\lambda }) = \{ {\lambda }_i - i + {c} + \tfrac{1}{2}, i \in \mathbb {Z}_{\ge 1} \} \end{aligned}$$

(B.6)

is a DPP when c is distributed according the discrete Gaussian distribution

$$\begin{aligned} \mathbb {P}(c) = \frac{t^cu^{c^2/2}}{\vartheta _3(t;u)}. \end{aligned}$$

(B.7)

Here, u is the same parameter as that of $\mathbb {P}_{u,\theta }^{m}$, but t can be chosen arbitrarily (it is related with the fermionic chemical potential). The normalization $\vartheta _3(t;u) := \sum _{c\in \mathbb {Z}} t^c u^{c^2/2}$ is a Jacobi theta function.

By [54, Theorem A] or [40, Theorem 3.1], the correlation kernel of $S_{{c}}({\lambda })$ reads explicitly

$$\begin{aligned} \mathcal {J}_{u,t,\theta }^m(k,\ell )&= \sum _{i \in \mathbb {Z}} \frac{t u^i}{1+t u^i}J_{k+i+\frac{1}{2}}(\Theta \gamma )J_{\ell +i+\frac{1}{2}}(\Theta \gamma ) \end{aligned}$$

(B.8)

$$\begin{aligned}&=\frac{1}{(2 \pi i)^2} \oiint _{c_+,c_-} \frac{\exp [\Theta S(z,k/\Theta )]}{\exp [\Theta S(w,\ell /\Theta )]} \cdot \frac{\kappa (z, w) dzdw}{wz}, \nonumber \\&\quad c_\pm : |z| = u^{\mp 1/4}, \nonumber \\ \kappa (z, w)&= \sum _{i \in \mathbb {Z}+\frac{1}{2}} \frac{t u^i}{1+t u^i} \left( \frac{z}{w}\right) ^i= \sqrt{\frac{w}{z}} \cdot \frac{(u;u)^2_{\infty }}{\vartheta _u(w/z)} \cdot \frac{\vartheta _3 (t z/w; u)}{\vartheta _3(t;u)}. \end{aligned}$$

(B.9)

using the notation $\vartheta _u(x):= (x; u)_\infty (u/x; u)_\infty$ and reusing the action notation for the order m multicritical measure defined at (3.10). The equivalence between the two forms of $\kappa$ is a special case of Ramanujan’s ${}_1\Psi _1$ summation [58], and the choice of contours with $|w|<|z|$ ensures the sum converges. Note the similarity with the integral expression for the zero temperature kernel (A.2). The proof of this in [40] adapts Okounkov’s fermionic approach (see Theorem 15) to the positive temperature setting, the $\kappa (z,w)$ given in (B.9) is the corresponding generating function $\langle c^\dagger (z)c(w)\rangle _{u,t} = \sum _{k,\ell } z^kw^{-\ell }\langle c^\dagger _kc_\ell \rangle _{u,t}$ of propagators.

The crossover regime The asymptotic regime of Theorem 19 is the one in which the “thermal” fluctuations coming from the factor of $u^{|\mu |}$ match the order of magnitude of the “quantum” fluctuations coming from the skew Schur functions, so that $\alpha$ parametrises a crossover between regimes where either kind of fluctuation dominate. Heuristically, from the identification $u =e^{-1/T}$ where T is the (dimensionless) temperature, the thermal fluctuations are of order of T, so comparing with scale of the fluctuations in the zero temperature case (i.e., the multicritical Schur measure) we look for a regime in which

$$\begin{aligned} T \sim \Theta ^{\frac{1}{2m+1}}. \end{aligned}$$

(B.10)

Fixing a specific regime

$$\begin{aligned} u:= \exp \left[ -\alpha (d \Theta )^{-\frac{1}{2m+1}}\right] , \qquad \theta := \alpha d^{-\frac{1}{2m+1}}\Theta ^{\frac{2m}{2m+1}} \end{aligned}$$

(B.11)

by this reasoning, it is straightforward to see that it is asymptotically equivalent to the crossover regime in the statement.

Proof of Theorem 19

Our proof follows that of [40], with some adaptations that correspond precisely to the arguments of Sect. 3.3 of this text. It consists of three steps.

(i) Shift-mixing

Let $(\lambda ,\mu )$ be distributed according to $\mathbb {P}_{u,\theta }^{m}$, and c distributed according to (B.7) with $t=1$. Then, by the determinantal nature of the shift-mixed process (B.6), we have

$$\begin{aligned} \mathbb {P}({\lambda }_1 + {c} < \ell _s ) = \det (1 - \mathcal {J}_{u,1,\theta }^m )_{l^2(\ell _s + \mathbb {Z}_{\ge 0})} \end{aligned}$$

(B.12)

with $\ell _s := \lfloor b\Theta + (d\Theta )^{\frac{1}{2\,m+1}}\rfloor - \frac{1}{2}$ and $\mathcal {J}_{u,1,\theta }^m$ the correlation kernel (B.8) specialized at $t=1$. Our task is now to analyze the Fredholm determinant in the asymptotic regime of the theorem, where $\theta \rightarrow \infty$, $u \rightarrow 1$ with $\theta (1-u)^{2\,m} \rightarrow \alpha ^{2\,m+1}$ (and $\Theta := \theta /(1-u)$).

(ii) Asymptotic analysis Let us start from the integrand of $\mathcal {J}_{u,1,\theta }^m(k,\ell )$ in a regime where $k = \lfloor b\Theta + x(d\Theta )^{1/(2\,m+1)} \rfloor - \tfrac{1}{2}$ and $\ell = \lfloor b\Theta + y(d\Theta )^{1/(2m+1)} \rfloor - \tfrac{1}{2}$, which is (suppressing floor functions)

$$\begin{aligned} \frac{ \exp \left[ \Theta S(z;b) - x (d\Theta )^{\frac{1}{2m+1}}\log z\right] }{\exp \left[ \Theta S(w;b) - y (d\Theta )^{\frac{1}{2m+1}}\log w\right] }\cdot \kappa (z, w). \end{aligned}$$

(B.13)

Since $\Theta \rightarrow \infty$ in our asymptotic regime, we can directly use $\Theta$ as a large parameter, and then for everything except for the function $\kappa (z,w)$, the steepest descent analysis follows precisely the arguments of Sect. 3.3 (with just a change from $\theta$ to $\Theta$). At $z=w=1$ there is an order 2m saddle point, and we use the same change of variables

$$\begin{aligned} z = \exp \left[ \zeta (d \Theta )^{-\frac{1}{2m+1}} \right] , \quad w = \exp \left[ \omega (d \Theta )^{-\frac{1}{2m+1}} \right] . \end{aligned}$$

(B.14)

The arguments for the tails bound generalise. The contour $c_+$ of the integral in z is circle on which

$$\begin{aligned} |z| = u^{-1/4} = \exp [\Re (\zeta )(d \Theta )^{-{1}/{(2m+1)}}], \end{aligned}$$

(B.15)

and as $u \rightarrow 1$ this is satisfied if and $\Re (\zeta ) \sim (d\Theta )^{1/(2\,m+1)}/4 (1-u) \sim \alpha /4$, so the central region is asymptotically parametrised by $\zeta \in i \mathbb {R}+\alpha /4$ and $\omega = i \mathbb {R}-\alpha /4$.

At the same time, $\kappa$ has a reasonable asymptotic behaviour in the above regime and on the contours $c_\pm$. First, when z, w are around around 1, observing that $z = u^{-\zeta /\alpha }, w = u^{-\omega /\alpha }$, we have

$$\begin{aligned} \kappa (z, w) = \sum _{i \in \mathbb {Z}+\frac{1}{2}} \frac{(z/w)^i}{1+u^{-i}} \sim \alpha (d \Theta )^{-\frac{1}{2m+1}} \cdot \frac{\pi }{\sin \frac{\pi (\zeta - \omega )}{\alpha }} \quad \text {as } \Theta \rightarrow \infty . \end{aligned}$$

(B.16)

This follows by the same argument as that leading to [40, Equation (5.32)]: putting $u = e^{-r}$ and $z/w = e^{r/2+i\phi }$ for $\phi \in [-\pi , \pi ]$, by the Poisson summation formula we have

$$\begin{aligned} \kappa (z,w)&= \sum _{k \in \mathbb {Z}+ \frac{1}{2}} \frac{e^{i\phi k}}{2 \cosh \frac{rk}{2}} = \sum _{n \in \mathbb {Z}} (-1)^n \frac{\pi }{r\cosh \frac{\pi (\phi - 2 \pi n)}{r}} \end{aligned}$$

(B.17)

and on the contours $c_{\pm }$⁷ as $\Theta \rightarrow \infty , u \rightarrow 1$,

$$\begin{aligned} \kappa (z,w) \sim \frac{\pi }{r\cosh \frac{\pi \Im (\zeta - \omega )}{\alpha }} = \frac{\pi }{r\sin \frac{\pi (\zeta - \omega )}{\alpha }} \end{aligned}$$

(B.18)

The prefactor $(d \Theta )^{-\frac{1}{2m+1}}$ will be cancelled by part of the Jacobian for the change of variables $(z, w) \mapsto (\zeta , \omega )$. From the same Poisson summation formula, we see that outside of the central region around $z = w = 1$, $\kappa$ decays exponentially fast to 0, see [40, Lemma 5.5].

Putting everything together and noting that the same exponential decay bounds imply dominated convergence, as $\Theta \rightarrow \infty$ and $u \rightarrow 1$ we have

$$\begin{aligned}&(d \Theta )^{\frac{1}{2m+1}} \mathcal {J}_{u,1,\theta }^m \left( \lfloor b\Theta + x(d\Theta )^{1/(2m+1)} \rfloor - \tfrac{1}{2}, \lfloor b\Theta + y(d\Theta )^{1/(2m+1)} \rfloor - \tfrac{1}{2} \right) \nonumber \\&\quad \rightarrow \frac{1}{(2\pi i)^2}\int _{ i \mathbb {R}+\frac{\alpha }{4}} \int _{ i \mathbb {R}-\frac{\alpha }{4}} \frac{\exp \left[ (-1)^{m+1}\frac{ \zeta ^{2m+1}}{2m+1} - x \zeta \right] }{\exp \left[ (-1)^{m+1} \frac{\omega ^{2m+1}}{2m+1} - y \omega \right] } \cdot \frac{\pi }{\alpha \sin \frac{\pi (\zeta -\omega )}{\alpha }} d\omega d\zeta . \end{aligned}$$

(B.19)

Using the identity

$$\begin{aligned} \frac{\pi }{\alpha \sin \frac{\pi (\zeta -\omega )}{\alpha }} = \int _{-\infty }^\infty \frac{e^{(\alpha +\omega -\zeta ) v} d v}{1+e^{\alpha v}} \end{aligned}$$

(B.20)

valid for $0<\Re (\zeta -\omega )<\alpha$, we see that the limiting kernel is equal to $\mathcal {A}^\alpha _{2m+1}(x, y)$.

The same exponential decay arguments for the integrand apply again to the integral, so the traces of $\mathcal {J}_{u,1,\theta }^m$ also converges uniformly to the traces of $\mathcal {A}^\alpha _{2m+1}$ on any set that is bounded below. Since the Hadamard bound argument equally applies here, we have convergence of the Fredholm determinants too, with

$$\begin{aligned} \mathbb {P}\left[ \frac{{\lambda }_1 + {c} - b\Theta }{(d\Theta )^{\frac{1}{2m+1}}} < s\right] \rightarrow \det (1- \mathcal {A}_{2m+1}^\alpha )_{L^2([s,\infty ))}. \end{aligned}$$

(B.21)

(iii) Shift removal The limiting distribution (B.21) above is not quite what we wanted to prove due to the random shift c. Luckily it can be removed without affecting the result: indeed, by [40, Lemma 2.1], ${c} / \Theta ^{1/(2m+1)}$ converges to 0 in probability (recall that we set $t=1$ here). $\square$

C. Generalised Higher-Order Airy Kernel

In this appendix we extend the multicritical measures to have more general asymptotic edge distributions of a kind shown by Cafasso, Claeys and Girotti [7] to encode Fredholm determinant solutions of the general Painlevé II hierarchy. The authors found that if we set

$$\begin{aligned} p_{\tau ;2m+1}(x):= \frac{x^{2m+1}}{2m+1} + \sum _{i=1}^{m-1} \frac{\tau _i}{2i+1} x^{2i+1} \end{aligned}$$

(C.1)

for a given sequence of $m-1$ real constants $\tau = (\tau _1,\ldots ,\tau _{m-1})$, then the Fredholm determinant

$$\begin{aligned} F_{\tau ;2m+1}(s) = \det (1-\mathcal {A}_{\tau ;2m+1})_{L^2([s,\infty ))} \end{aligned}$$

(C.2)

of the generalised higher-order Airy kernel

$$\begin{aligned} \mathcal {A}_{\tau ;2m+1}(x,y)= & {} \frac{1}{(2\pi i)^2} \int _{i \mathbb {R}+1} \int _{ i \mathbb {R}-1} \frac{\exp [(-1)^{m+1}p_{\tau ;2m+1}(\zeta )-x\zeta ]}{ \exp [(-1)^{m+1}p_{\tau ;2m+1}(\omega )-y\omega ]}\nonumber \\{} & {} \quad \frac{d \zeta d \omega }{\zeta -\omega }. \end{aligned}$$

(C.3)

is related to a solution $q_{\tau ;m}(s)$ of the order 2m general Painlevé II hierarchy equation with coefficients $\tau _i$ by

$$\begin{aligned} F_{\tau ;2m+1}(s) = \exp \left[ - \int _s^\infty (x-s) q_{\tau ;m}^2 ((-1)^{m+1}x) \, dx\right] . \end{aligned}$$

(C.4)

This relation generalises (1.11), which corresponds to the case $\tau = (0,0,\ldots )$.

Generalised Multicritical Fermions and Schur Measures The generalised higher-order Airy functions

$$\begin{aligned} {{\,\textrm{Ai}\,}}_{\tau ;2m+1}(x) = \frac{1}{2\pi i} \int _{i\mathbb {R}+1} \exp [(-1)^{m+1}p_{\tau ;2m+1}(\zeta )-x\zeta ] d\zeta , \end{aligned}$$

(C.5)

making up the kernel $\mathcal {A}_{\tau ;2m+1}$ satisfy the eigenfunction relations

$$\begin{aligned} (-1)^{m+1} \left[ \frac{d^{2m}}{dx^{2m}} + \sum _{i=1}^{m-1} \tau _i \frac{d^{2i}}{dx^{2i}} \right] {{\,\textrm{Ai}\,}}_{\tau ;2m+1}(x) = -x {{\,\textrm{Ai}\,}}_{\tau ;2m+1}(x), \end{aligned}$$

(C.6)

generalising (2.19). One can adapt the flat trap models of [6] to recover momentum space edge Hamiltonians of the above form, generalising (2.18). This can be achieved for instance by considering trapping potentials of the form

$$\begin{aligned} V(x) = x^{2m} + \sum _i (-1)^i \tau _i p_{\text {edge}}^{\frac{2m - 2i}{2m+1}}x^{2i} \end{aligned}$$

(C.7)

with the same scaling regime $p_{\text {edge}}\rightarrow \infty$ as that considered in Sect. 2.2; note that finer tuning is required than in the $\tau = (0,0,\ldots )$ case. We focus on a discrete construction, which coincides with the momentum space edge of such a model in a suitable continuum limit. Our main task is to identify the correct asymptotic regime.

We again construct Hermitian Schur measures (and corresponding lattice fermion models) with a single real parameter $\theta$, but no longer require each Miwa time in the Schur function specialisation to grow linearly with $\theta$; once we consider combinations of Miwa times growing at different speeds, we can tune the speeds so that the integrand of the limiting edge kernel has a given odd polynomial in the exponential, from the same saddle point analysis of Sect. 3.3.

To be specific, we combine the coefficients $\gamma _r$ already used to define multicritical measures, to define generalised ones as follows (where we emphasise that the sequence of constants $\gamma$ is replaced with a $\theta$-dependant functions $\gamma ^{\tau }(\theta )/\theta$):

Definition 20

(Generalised multicritical measure) Fix a sequence of $m-1$ real constants $\tau = (\tau _1,\ldots ,\tau _{m-1})$, and choose m sequences of real coefficients $\gamma ^{(1)}, \ldots , \gamma ^{(m)}$ where $\gamma ^{(i)}$ satisfies the conditions for an order i multicritical measure and has right edge and fluctuation coefficients $b_i,d_i$. Then, for a positive parameter $\theta$, we define the sequence $\gamma ^{\tau }(\theta )$ of Miwa times, with elements indexed $r \ge 1$

$$\begin{aligned} \gamma ^{\tau }(\theta )_r = \theta \gamma _r^{(m)} + \sum _{i=1}^{m-1} \theta ^{\frac{2i+1}{2m+1}} (-1)^{m-i} \frac{\tau _i}{d_i} \gamma _r^{(i)} \end{aligned}$$

(C.8)

and we define an order m generalised multicritical measure

$$\begin{aligned} \mathbb {P}^{\tau ;m}_\theta (\lambda ) = \frac{1}{Z} s_{\lambda }[\gamma ^{\tau }(\theta )]^{2}, \qquad Z = e^{\sum _{r} r \gamma ^{\tau }(\theta )_r^2} \end{aligned}$$

(C.9)

along with its edge position function

$$\begin{aligned} B(\theta ) = b_m \theta + \sum _{i=1}^{m-1} b_i (-1)^{m-i} \frac{\tau _i}{d_i} \theta ^{\frac{2i+1}{2m+1}}. \end{aligned}$$

(C.10)

This generalisation is defined so that we have the edge behaviour we would expect in analogy to Theorem 1:

Theorem 21

(Edge fluctuations in generalised multicritical measures) If ${\lambda }$ is a random partition under the generalised multicritical measure $\mathbb {P}^{\tau ;m}_\theta (\lambda )$, then we have

$$\begin{aligned} \lim _{\theta \rightarrow \infty } \mathbb {P}^{\tau ;m}_{\theta } \left[ \frac{{\lambda }_1 - B(\theta ) }{(d_m\theta )^{\frac{1}{2m+1}}} \le s \right] = \det (1 - {\mathcal {A}}_{\tau ;2m+1})_{L^2(s, \infty )} =: {F}_{\tau ;2m+1}( s).\nonumber \\ \end{aligned}$$

(C.11)

It is worth highlighting that the expected edge position $B(\theta )$ now has quite a non-trivial expansion: it has deterministic terms of orders $\theta$, $\theta ^{\frac{2n-1}{2n+1}}$, $\dots$, $\theta ^{\frac{3}{2n+1}}$, and only at order $\theta ^{\frac{1}{2n+1}}$ do we encounter the fluctuations. The expected size is also more subtle: since we have ${\mathbb {E}}(|{\lambda }|) = \sum _{r\ge 1} r^2 \gamma (\theta )_r^2$, only the leading order term now scales with $\theta ^2$.

Tuning speeds and coefficients The proof of Theorem 21 involves no new arguments than the ones of Sect. 3.3, so we find it more instructive to present an informal derivation of Definition 20. To do so, let us define additional notation, putting

$$\begin{aligned} S^{(i)}(z;x)\!=\!\sum _{r\ge 1}\!\gamma _r^{(i)}\!\big (z^r - z^{-r}\big )\!-x\log z \!=\! V^{(i)}(z) \!-\! V^{(i)}(z^{-1}) \!-\! x\log z \end{aligned}$$

(C.12)

for the action and potential associated with the coefficients $\gamma ^{(i)}$. Since each $\gamma ^{(i)}$ defines an order i multicritical measure with right edge and fluctuation coefficients $b_i, d_i$, we have, by (3.14) and (3.15), the following expansion of $S^{(i)}$ around $z=1$:

$$\begin{aligned} S^{(i)}(z;b_i) = \frac{(-1)^{i+1}d_i}{2i+1} (z-1)^{2i+1} + O((z-1)^{2i+3}). \end{aligned}$$

(C.13)

Let us form a generalised potential, which now scales with $\theta$,

$$\begin{aligned} {\textsf{V}}(z) = \sum _{i=1}^m f_i(\theta ) V^{(i)}(z); \end{aligned}$$

(C.14)

we fix $f_m(\theta ) = 1$ for convenience. Our goal is now to find suitable $f_i(\theta )$ so as to obtain the scaling regime of Theorem 21 and the limiting edge kernel $\mathcal {A}_{\tau ;2m+1}$. We will just look at the integrand in the double contour integral representation in a region near the multicritical saddle point. The discrete kernel we start with is

$$\begin{aligned} \mathcal {J}_\theta ^{\tau ;m}(k, \ell ) = \frac{1}{(2 \pi i )^2}\oiint _{c_+,c_-} \frac{ \exp [\theta ( {\textsf{V}}(z)-{\textsf{V}}(z^{-1})) ]}{ \exp [\theta ({\textsf{V}}(w)-{\textsf{V}}(w^{-1}) )]} \frac{dz dw}{z^{k+\frac{1}{2}} w^{-\ell +\frac{1}{2}} (z-w)}\nonumber \\ \end{aligned}$$

(C.15)

for $k,\ell \in \mathbb {Z}+\frac{1}{2}$, with $c_+$ for the integration in z passing just outside the unit circle and $c_-$ for w passing just inside. Now we set

$$\begin{aligned} {\textsf{S}}(z;x) = {\textsf{V}}(z)- {\textsf{V}}(z^{-1}) - x\log z; \quad {\textsf{b}}(\theta ):= \sum _{i=1}^m f_i(\theta )b_i. \end{aligned}$$

(C.16)

Then, if we rewrite the coordinates relative to $k={\textsf{b}}(\theta )+k', \ell ={\textsf{b}}(\theta )+\ell '$ the kernel may be written

$$\begin{aligned} \mathcal {J}_\theta ^{\tau ;m}(k, \ell ) = \frac{1}{(2 \pi i )^2}\oiint _{c_+,c_-}\frac{\exp [\theta (\textsf{S}(z;\textsf{b}(\theta ))-\textsf{S}(w;\textsf{b}(\theta )))] dz dw}{z^{k'+1/2} w^{-\ell '+1/2} (z-w)} \end{aligned}$$

(C.17)

Since we have

$$\begin{aligned} {\textsf{S}}(z;{\textsf{b}}(\theta )) = \sum _{i=1}^m f_i(\theta ) S^{(i)}(z;b_i), \end{aligned}$$

(C.18)

near the order 2i saddle point for each $S^{(i)}$, we let $\varepsilon$ be a small positive number that tends to zero as $\theta$ tends to infinity and consider a change of variables

$$\begin{aligned} z = 1 + \zeta \varepsilon , \qquad w = 1 + \omega \varepsilon , \qquad k'=\frac{x}{\varepsilon }, \qquad \ell '=\frac{y}{\varepsilon } \end{aligned}$$

(C.19)

(this simple setup is sufficient for our purposes; we will parametrise the contours explicitly once we have suitable $\varepsilon$ and $f_i(\theta )$). Expanding in small $\varepsilon$ and using (C.13), the leading order approximation of the integrand is

$$\begin{aligned}{} & {} \frac{1}{\varepsilon (\zeta -\omega )} \exp \left[ \sum _{i=1}^m \theta f_i(\theta ) \frac{ (-1)^{i+1} d_i}{2i+1} \varepsilon ^{2i+1}(\zeta ^{2i+1} \right. \nonumber \\{} & {} \left. - \omega ^{2i+1}) - x \zeta + y \omega + O(\theta \varepsilon ^{2m+3})\right] . \end{aligned}$$

(C.20)

It now becomes clear that in the generalised multicritical action, each $f_i(\theta )$ should scale as $\varepsilon ^{-2i-1}/\theta$. More precisely, to use our convention that $f_m(\theta )= 1$, we identify $\varepsilon = (d_m\theta )^{-1/(2m+1)}$ (which indeed tends to 0) to be the appropriate scale; taking an action with

$$\begin{aligned} f_i(\theta ):= (-1)^{m-i} \frac{\tau _i}{d_i} \theta ^{\frac{2i-2m}{2m+1}}, \quad i=1,\dots ,m-1, \end{aligned}$$

(C.21)

the leading order term coincides precisely with the integrand of $\mathcal {A}_{\tau ,2m+1}$. At the level of the parametrised specialisations for the corresponding Schur measures, this gives corresponds precisely to Miwa times $\gamma ^\tau (\theta )_r$ corresponding the generalised multicritical measure $\mathbb {P}^{\tau ;m}_\theta$. The function ${\textsf{b}}(\theta )$ determining the edge scaling becomes $B(\theta )$ defined in (C.10).

The edge asymptotics With $f_i(\theta ), \varepsilon$ now determined, let us briefly discuss the remaining analysis needed to prove Theorem 21. From noting that the Jacobian for the change of variables from z, w to $\zeta , \omega$ contributes a factor of $\varepsilon ^2$, we see that $(d_m\theta )^{1/(2\,m+1)}\mathcal {J}_\theta ^{\tau ;m}$ is the relevant rescaled kernel.

Comparing to the analysis of Sect. 3.3, note that the tails bound and the exponential decay apply immediately to this case. The same contours can be reused along with the same dominated convergence arguments, to show firstly the uniform convergence

$$\begin{aligned}&(d_m\theta )^{\frac{1}{2m+1}}\mathcal {J}_\theta ^{\tau ;m} (\lfloor B(\theta ) + x (d_m\theta )^{\frac{1}{2m+1}}\rfloor -\tfrac{1}{2} ,\lfloor B(\theta ) + y (d_m\theta )^{\frac{1}{2m+1}}\rfloor -\tfrac{1}{2}) \nonumber \\&\quad \rightarrow \mathcal {A}_{\tau ;2m+1}(x,y) \end{aligned}$$

(C.22)

as $\theta \rightarrow \infty$, and in turn the convergence of traces and finally of Fredholm determinants uniformly on sets bounded below, with

$$\begin{aligned} \lim _{\theta \rightarrow \infty }\mathbb {P}\left[ \frac{{\lambda }_1 - B(\theta )}{(d_m\theta )^{\frac{1}{2m+1}}} < s\right]&= \lim _{\theta \rightarrow \infty }\det (1-\mathcal {J}_\theta ^{\tau ;m})_{l^2\left( \mathbb {Z}_{\ge 0}+ \lfloor B(\theta ) + s(d_m\theta )^{\frac{1}{2m+1}}\rfloor -\frac{1}{2} \right) }\nonumber \\&= \det (1- \mathcal {A}_{\tau ;2m+1})_{L^2([s,\infty ))} \end{aligned}$$

(C.23)

as required.

Finally, let us note that the extensions presented in this appendix and in Appendix 1 are completely compatible; we can directly construct analogous “generalised cylindric multicritical measures” using the Miwa time specialisations of Definition 20. The distributions $F^\alpha _{2m+1}$ then generalise to Fredholm determinants of positive temperature kernels composed of the functions ${{\,\textrm{Ai}\,}}_{\tau ;2m+1}$.