#  Multidimensional pseudo Leja sequences

Dimitri Jordan Kenne  Doctoral school of Exact and Natural Sciences,
Jagiellonian University, Łojasiewicza 11, Kraków, 30-348, Lesser Poland,
Poland<EMAIL_ADDRESS>
(Date: December 2023)

###### Abstract.

The one-dimensional pseudo Leja sequences introduced in [5], as an alternative
to Leja sequences, provides us with good interpolation nodes for the
approximation of holomorphic functions. We propose a definition of
multidimensional pseudo Leja sequences associated to a compact set $K$ of the
complex space $\mathbb{C}^{p}$ which generalises both the one-dimensional
version and the multidimensional Leja sequences. We show that these sequences
can be used to calculate the transfinite diameter of $K$. We also present a
relation to the pluricomplex Green function associated to $K$. Subsequently,
we show that the intertwinning of pseudo Leja sequences is still a pseudo Leja
sequence. We give a method to compute pseudo Leja sequences with the help of
discrete meshes.

###### Key words and phrases:

Lagrange interpolation, Leja sequences, transfinite diameter, Green function,
admissible mesh, intertwining sequence

###### 2000 Mathematics Subject Classification:

Primary 32E30, 41A05, 41A63

##  1\. Introduction

We are interested in the computation of goods points for approximation by
multivariate polynomial interpolation. Here, by good points, we mean arrays of
interpolation nodes in a regular ( see below) compact set $K$ for which the
corresponding sequence of Lagrange interpolation polynomials of any
holomorphic function $f$ on a neighbourhood of $K$, converges uniformly to $f$
on $K$. In the univariate case, a characterisation of such points is very
well-know ([7], [8], [18]). In contrast, in the multivariate case, no
characterisation is known and very few examples of such good points are
currently available. The only general example is given by arrays of Fekete
points (Fekete points for $K\subset\mathbb{C}^{p}$ are points which maximises
the (modulus) of the Vandermonde determinant of a given order, see below).
Unfortunately, these points are impossible to determine explicitly and even
the numerical computation of their discrete version is complicated, so that
they are essentially of theoretical interest. Apart from the Fekete points,
the sole examples of good points are furnished by the intertwining of
univariate good points ([5], [4], [23]) and special remarkable configurations
such as the famous Padua points in the unit square of $\mathbb{R}^{2}$ ( [10],
[13], [14]). The interest of other configurations is supported by numerical
experiments (see e.g. [1], [12]).

In the univariate case, while the nature of good points is completely
understood from the theoretical point of view, the computation of such points
for an arbitrary regular compact set is far from being easy. A way of
constructing univariate good points were introduced in [5] with the notion of
pseudo Leja sequences. Such a sequence is defined recursively as follows. The
first point $\xi_{0}\in K$ is chosen arbitrarily and, if the first $j$ points
$\xi_{0},\dots,\xi_{j-1}$ are already obtained, then $\xi_{j}$ is chosen in
$K$ to ensure that

(1) $M_{j}\prod_{i=0}^{j-1}|\xi_{j}-\xi_{i}|\geq\max_{z\in
K}\prod_{i=0}^{j-1}|z-\xi_{i}|.$

Here $(M_{j})$ is a sequence of real numbers $M_{j}\geq 1,\ j\geq 1$ which is
required to be of sub-exponential growth. The classical Leja points correspond
to the special case, $M_{j}=1$, $j\geq 1$. The point is that the possibility
$M_{j}>1$ alleviates the computation of the points without weakening the
approximation quality of the Leja points : it is shown in [5, Theorem 1, page
56] that they are good points (for reasonable compact sets) in the sense
above. The strong result that, on a set of positive logarithmic capacity, the
Lebesgue constants of one-dimensional pseudo Leja points with bounded sequence
$(M_{j})$ are subexponential has been recently established in [24].

It is readily seen that the product in (1) can be replaced by the (modulus) of
a Vandermonde determinant and this gives the way to the extension of the
definition to the higher dimensional case. The study of this generalisation is
the subject of the present paper. The precise definition of a multivariate
pseudo Leja sequence is given in Section 2 . Although we are not able to show
that our multivariate pseudo Leja points are good points (in the multivariate
case, this is not even known for ordinary Leja sequence), we prove that they
permit to recapture the multivariate transfinite diameter of $K$, which, in
turn, by deep results due to Berman, Boucksom and Nyström ([3, Theorem A, page
3 ]) and Bloom ([9, Corollary 4.5, page 1563]), implies that they permit to
recover the multivariate equilibrium measure and the pluri-complex Green
function for a regular compact set $K$. This is usually regarded as an
indicator that the points should actually be good in the above sense,
especially because these conditions are equivalent in the univariate case (see
[18, Theorem 1, page 65] or [8, Theorem 1.5, page 446]).

In Section5, we will briefly explain how to compute pseudo-Leja points,
showing that discrete Leja points can be seen as examples of pseudo-Leja
points.

We will also prove an interesting stability property of multivariate pseudo
Leja sequences. Roughly speaking, by intertwining, in some specific way, the
points of a pseudo Leja sequence for $K_{1}\in\mathbb{C}^{p_{1}}$ and that of
a pseudo Leja sequence for $K_{2}\in\mathbb{C}^{p_{2}}$, we obtain a pseudo
Leja sequence for the product $K_{1}\times K_{2}$, see 4.

For the convenience of the reader, we will now recall standard notations and
basic definitions on multivariate interpolation. Multi-indices will be ordered
according to the graded lexicographic order, so that $\alpha\prec\beta$ if
$|\alpha|<|\beta|$ or $|\alpha|=|\beta|$ and the left-most non zero element of
$\alpha-\beta$ is negative. We arrange the elements of $\mathbb{N}^{p}$ in an
increasing sequence $\kappa=\kappa^{(p)}$ (with respect to $\prec$),

(2) $\kappa=\kappa^{(p)}:\mathbb{N}\to\mathbb{N}^{p}.$

Thus, when $p=2$, we have

$\kappa(0)=(0,0)\prec\kappa(1)=(0,1)\prec\kappa(2)=(1,0)\prec\kappa(3)=(0,2)\\\
\prec\kappa(4)=(1,1)\prec\kappa(5)=(2,0)\prec\kappa(6)=(0,3)\prec\dots.$

The monomials of $p$ complex variables will be denoted as follows

(3) $e_{N}(z)=z^{\kappa(N)}=z_{1}^{\alpha_{1}}\cdots
z_{p}^{\alpha_{p}},\quad\kappa(N)=(\alpha_{1},\dots,\alpha_{p}).$

We will use the space of polynomials spanned by the first $N$ monomials :

$\mathcal{P}_{N}=\textbf{span}\\{e_{i}:\ i=0,\dots,N-1\\},\quad N\geq 1.$

The dimension of the classical space of polynomial of total degree not bigger
than $d$ is of dimension

$h_{d}=\binom{p+d}{d},$

thus

$N=h_{d}-1\implies\mathcal{P}_{N}=\textbf{span}\\{z\to
z^{\alpha}\;:\;|\alpha|\leq d\\}.$

The Vandermonde determinant of a set of the points $\xi_{0},\dots,\xi_{q}$ in
$\mathbb{C}^{p}$ is defined as

(4)
$\textsc{vdm}(\xi_{0},\dots,\xi_{q})=\det\left[e_{N}(\xi_{M})\right]_{0\leq
M,N\leq q}.$

In particular, $\textsc{vdm}(\xi_{0})=1$. Observe that its modulus is
independent of the ordering of the points. The Vandermonde determinant is a
polynomial in the coordinates of the points $\xi_{j}$, thus a polynomial in
$p(q+1)$ variables. As such, its degree for $q=h_{d}-1$ is given by

(5) $l_{d}=\sum_{i=1}^{d}i(h_{i}-h_{i-1})=p\binom{p+d}{p+1}.$

Given a set of $N$ points,
$\Omega_{N}=\\{\zeta_{N0},\dots,\zeta_{NN}\\}\subset\mathbb{C}^{p}$, with a
non-zero Vandermonde determinant, we can form the Fundamental Lagrange
Interpolation Polynomials (FLIP)

(6)
$l_{j}^{(N)}(z)=\dfrac{\textsc{vdm}(\zeta_{N0},\dots,\zeta_{N(j-1)},z,\zeta_{N(j+1)}\dots,\zeta_{NN})}{\textsc{vdm}(\zeta_{N0},\dots,\zeta_{NN})},\quad
j=0,\dots,N.$

Observe that $l_{j}^{(N)}$ is an element of $\mathcal{P}_{N+1}$. For a
function $f$ defined at the points in $\Omega_{N}$,

(7) $L_{\Omega_{N}}f(z)=\sum_{j=0}^{N}f(\zeta_{Nj})l_{j}^{(N)}(z)$

is the Lagrange Interpolation Polynomial (LIP) of $f$ in $\mathcal{P}_{N+1}$
at the points of $\Omega_{N}$. In particular,

(8) $f\in\mathcal{P}_{N+1}\implies f=L_{\Omega_{N}}f(z).$

In the whole paper the compact sets $K$ considered will be assumed to be
(polynomially) determining. This means that no nonzero polynomial vanishes on
$K$. Equivalently, $K$ is not included in an algebraic set. This property can
also be expressed in terms of Vandermonde determinants. Namely, setting

(9) $V_{j}(K)=\max_{\xi_{0},\dots,\xi_{j-1}\in
K}|\textsc{vdm}(\xi_{0},\dots,\xi_{j-1})|,\quad j\geq 1,$

then $K$ is determining if and only if no $V_{j}(K)$ vanishes. The sufficiency
follows from an application of (8). To see the converse, suppose
$V_{N-1}(K)\not=0$ for $1\leq j<N$ but $V_{N}(K)=0$ and let
$\\{\xi_{0},\dots\xi_{N-1}\\}\subset K$ such that
$\textsc{vdm}(\xi_{0},\dots\xi_{N-1})\not=0$. The polynomial
$Q_{N}(z)=\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},z)$ vanishes on $K$ but is not
the zero polynomial since its $z^{\kappa(N)}$ coefficient is not null so that
$K$ is not determining.

If it is necessary to specify the dimension of the ambient space, we will add
a superscript to the above objects thus writing for instance $k^{(p)}(N)$,
$\mathcal{P}_{N}^{(p)}$, and so on.

For survey on the actual state of knowledge on multivariate polynomial
interpolation we refer to [7].

##  2\. Pseudo Leja sequences

######  Definition 2.1.

Let $K$ be a determining compact subset in $\mathbb{C}^{p}$. On says that
$(\xi_{j})$ is a pseudo Leja sequence in $K$ if there exists a sequence of
real numbers $(M_{j})_{j\geq 1}\subset[1,+\infty)$ satisfying:

(10) $M_{j}|\textsc{vdm}(\xi_{0},\dots,\xi_{j-1},\xi_{j})|\geq\max_{z\in
K}|\textsc{vdm}(\xi_{0},\dots,\xi_{j-1},z)|\quad\text{for any $j\geq 1$,}$

with the growth condition,

(11) $\lim_{d\to+\infty}(\max_{h_{d-1}\leq j<h_{d}}M_{j})^{1/d}=1.$

We say that $\mathcal{L}$ is a pseudo-Leja sequence of Edrei growth
$(M_{j})_{j\geq 1}$.

In the case $p=1$, this definition coincides with that given in [5]. A Leja
sequence is a pseudo Leja sequence of Edrei growth $1$. The fact that $K$ is
determining immediately implies the existence of pseudo Leja sequence with non
vanishing Vandermonde determinant. Observe that we may with very limited loss
of generality, assume that $(M_{j})$ is non-decreasing and, in that case, (11)
reduces to a somewhat less clumsy condition.

Observe that if $(\xi_{j})$ is a pseudo Leja sequence for $K$ then

(12) $|\textsc{vdm}(\xi_{0},\dots,\xi_{j-1},\xi_{j})|>0,\quad\text{for all
$j$}.$

This follows from the same reasoning sketched in the introduction : if $j$ was
the smallest integer for which $|\textsc{vdm}(\xi_{0},\dots,\xi_{j-1},z)|=0$
on $K$ then $P(z)=\textsc{vdm}(\xi_{0},\dots,\xi_{j-1},z)$ would be a non zero
polynomial vanishing on $K$ thus contradicting the fact that it is
determining. The non vanishing of the Vandermonde determinant, implies, see
(7), the following proposition.

######  Proposition 2.1.

Let $(\xi_{i})$ a pseudo Leja sequence as above. Lagrange interpolation at
$\Omega_{N}=\\{\xi_{0},\dots,\xi_{N}\\}$, $N\in\mathbb{N}$, is always
possible.

We immediately draw of a consequence of the growth condition that will be used
in the proof of our main theorem below.

######  Lemma 2.2.

If $\\{M_{j}\\}_{j\geq 1}$ is a sequence of real numbers satisfying Property
(11) from the definition above, then

$\lim_{d\to+\infty}\big{[}\prod_{j=1}^{h_{d}-1}M_{j}\big{]}^{1/l_{d}}=1,$

where $l_{d}$ is defined in (5).

###### Proof.

Notice that

$1\leq\prod_{j=1}^{h_{d}-1}M_{j}\leq(\max_{j=1,\dots,h_{d}-1}M_{j})^{h_{d}}\quad\text{and}\quad\frac{h_{d}}{l_{d}}=\frac{1}{d}\left(\frac{p+1}{p}\right).$

Choose the smallest $j(d)$ such that

$\max_{j=1,\dots,h_{d}-1}M_{j}=\max_{h_{j(d)-1}\leq j<h_{j(d)}}M_{j}.$

Since it is a non-decreasing sequence of integers, $(j(d))_{d\geq 1}$ tends to
$+\infty$ or is eventually constant. We examine both cases.

If $j(d)\to\infty$, then, by (11),

$\lim_{d\to+\infty}(\max_{h_{j(d)-1}\leq j<h_{j(d)}}M_{j})^{1/j(d)}=1,$

hence, since $0<j(d)\leq d$,

$\lim_{d\to+\infty}(\max_{h_{j(d)-1}\leq j<h_{j(d)}}M_{j})^{1/d}=1.$

If $j(d)$ is eventually constant, the conclusion is even simpler since
$(M_{j})$ is then bounded and $h_{d}/l_{d}\to 0$. ∎

We now state our main theoretical result and a consequence. A similar result
was obtained in [20] and [8] for Leja sequences. The method of proving Theorem
2.3 is in fact not different from that of Leja sequences except that here we
take into consideration the Edrei growth of the pseudo Leja sequence. Its
proof is postponed to the next section where we will recall the technical
points that we will need.

######  Theorem 2.3.

Let $K\subset\mathbb{C}^{p}$ be a determining compact set. If
$(\xi_{j})_{j\geq 0}\subset K$ is a pseudo-Leja sequence of Edrei growth
$(M_{i})_{j\geq 1}$, then

(13)
$\lim_{d\to+\infty}|\textsc{vdm}(\xi_{0},\dots,\xi_{h_{d}-1})|^{1/l_{d}}=D(K),$

where $D(K)$ denotes the multivariate transfinite diameter.

Thanks to R. Berman and S. Boucksom outstanding result which first appeared in
the preprint [2, Theorem 1.1] and later was presented as a consequence of a
more general convergence result in [3, Theorem A] with D. W. Nyström, we can
derive the following.

######  Theorem 2.4.

Let $K\subset\mathbb{C}^{p}$ be a determining compact set. If
$(\xi_{j})_{j\geq 0}\subset K$ is a pseudo-Leja sequence, then the normalised
counting measures $h_{d}^{-1}\sum_{j=0}^{h_{d}}[\xi_{j}]$ converge to the
equilibrium measure of $K$ in the weak-* sense.

Here $[\xi]$ refers to the Dirac measure at $\xi$ of total mass $1$.
Setting

(14)
$P_{k}(\xi)=\dfrac{\textsc{vdm}(\xi_{0},\dots,\xi_{k-1},\xi)}{\textsc{vdm}(\xi_{0},\dots,\xi_{k-1})},$

we will see that

(15)
$\lim_{d\to\infty}\big{(}\prod_{k=0}^{h_{d}-1}\|P_{k}\|_{K}\big{)}^{1/l_{d}}=D(K),$

and, by a strong result of Bloom ([6, Corollary 4.5]), we will obtain the
following.

######  Theorem 2.5.

Let $K$ be a polynomially convex regular compact subset of $\mathbb{C}^{p}$.
Then

(16)
$G_{K}(z)=\limsup_{N\to+\infty}\frac{1}{|\kappa(N)|}\log\left(\dfrac{|P_{N}(z)|}{\|P_{N}\|_{K}}\right),\quad
z\in\mathbb{C}^{p}\setminus K,$

where $G_{K}$ denotes the pluri-complex Green function of $K$.

For the notions of pluricomplex potential theory used in the above statement
(regularity, Green function, equilibrium measure), we refer the reader to
[21].

Observe that in the case where $K$ would be non determining, a pseudo Leja
sequence would be eventually arbitrary (as soon as $V_{j}(K)=0$), and Theorem
2.3 would be trivially true (with $D(K)=0$).

##  3\. Transfinite diameter and proof of Theorem 2.3

We will recall in details the objects and results related to the transfinite
diameter. We follow the presentation given in [8]. As usual, $K$ denotes a
determining compact set in $\mathbb{C}^{p}$. We will write $V_{j}$ instead of
$V_{j}(K)$, see (9), when there is no ambiguity. We define the $d$-th order
transfinite diameter of $K$, denoted by $D_{d}(K)=D_{d}$, as

$D_{d}=V_{h_{d}}^{1/l_{d}}.$

Fekete proved in [17] that the limit (as $d\to\infty)$ of the sequence
$D_{d}$, denoted as $D(K)$, exists for any compact set $K\subset\mathbb{C}$.
Later in [22], Leja introduced the name transfinite diameter and posed the
problem of its existence in the multivariate case. A positive answer to his
problem was given by Zaharjuta in the celebrated paper [25] where it also
shown a remarkable connection with some certain polynomials of minimal norm
that we will now describe.

A monic polynomial is a polynomial whose leading term with respect to $\prec$
is equal to $1$. Thus a monic polynomial of leading term $e_{i}$ is of the
form

(17) $P(z)=e_{i}(z)+\sum_{0\leq j<i}c_{j}e_{j}(z):\
c_{j}\in\mathbb{C}\\},\quad i\in\mathbb{N}.$

We will indicate such a form in writing $\textbf{lead}(P)=e_{i}$. We define
the $i$-th Chebyshev constant $\tau_{i}$ of $K$ as

(18)
$\tau_{i}=\inf\left\\{\|P\|_{K}^{1/|\kappa(i)|}\;:\;\textbf{lead}(P)=e_{i}\right\\}.$

Given $\theta=(\theta_{1},\dots,\theta_{p})$ in the standard $p$-simplex
$\Sigma$,

$\Sigma=\big{\\{}\theta\in\mathbb{R}^{n}\,:\,\sum_{i=1}^{p}\theta_{i}=1,\;\theta_{i}\geq
0,\;i=1,\dots,p\big{\\}},$

the limit

(19)
$\tau(K,\theta)=\limsup_{j\to+\infty,\frac{\kappa(j)}{|\kappa(j)|}\to\theta}\tau_{j}$

is called the directional Chebyshev constant of $K$ in the $\theta$-direction.
Now, if we denote by $\mathcal{T}_{d}$, the geometric mean of Chebyshev
constants for a given degree,

(20)
$\mathcal{T}_{d}=\big{[}\prod_{|\kappa(i)|=d}\tau_{i}\big{]}^{1/(h_{d}-h_{d-1})},$

the fundamental result of Zaharjuta can be stated as follows.

######  Theorem 3.1.

[25, Theorem 1] The sequence $\mathcal{T}_{d}$ converges as $d\to\infty$ and
we have

(21)
$\lim_{d\to\infty}\mathcal{T}_{d}=\exp\left[\frac{1}{meas(\Sigma)}\int_{\Sigma}\log(\tau(K,\theta))\
d\theta\right]=D(K)$

where $\text{meas}(\Sigma)$ denotes the standard volume of the simplex.

We will need a “mean modification” of the above.

######  Corollary 3.2.

We have

(22)
$\lim_{d\to\infty}\big{[}\prod_{j=1}^{d}\mathcal{T}_{j}^{r_{j}}\big{]}^{1/l_{d}}=D(K)$

where $r_{j}=j(h_{j}-h_{j-1})$ for $j\geq 1$.

The corollary follows from Theorem 3.1 by applying the following classical
lemma of Cesaro type (for which we omit the proof) with
$u_{d}=\log(\mathcal{T}_{d})$ and $\theta_{d,j}=r_{j}/l_{d}$.

######  Lemma 3.3.

Let $(u_{d})$ be a sequence of real numbers converging to $A$. Consider an
array of non negative numbers $\theta_{d,j}$ , $d,j\in\mathbb{N}^{\star}$
verifying

$\lim_{d\to+\infty}\max_{j=1.\dots,d}\theta_{d,j}=0\quad\text{and}\quad\sum_{j=1}^{d}\theta_{d,j}=1,\;d\in\mathbb{N}.$

Then the sequence

$s_{d}=\sum_{j=1}^{d}\theta_{d,j}u_{j},\ d\in\mathbb{N},$

also converges to $A$.

We have now assembled the material required in our proof of Theorem 2.3.

###### Proof of Theorem 2.3.

Set $L_{k}=|\textsc{vdm}(\xi_{0},\dots,\xi_{k-1})|$ for all $k\geq 1$. In view
of (9) and (12), we have

(23) $0<L_{k}\leq V_{k},\quad k\in\mathbb{N}.$

Recall that the polynomial $P_{k}$ defined in (14) is given by

(24)
$P_{k}(\xi)=\dfrac{\textsc{vdm}(\xi_{0},\dots,\xi_{k-1},\xi)}{\textsc{vdm}(\xi_{0},\dots,\xi_{k-1})}=e_{k}(\xi)+\sum_{0\leq
j<k}c_{j}e_{j}(\xi),$

Observe that

$P_{k}\in\mathcal{P}^{k},\quad\textbf{lead}{P_{k}}=e_{k},\quad\deg(P_{k})=\deg(e_{k})=\kappa(k).$

By definition of a pseudo Leja sequence, we have

$|P_{k}(\xi_{k})|=\dfrac{L_{k+1}}{L_{k}}\geq\dfrac{M_{k}^{-1}\max_{\xi\in
K}\textsc{vdm}(\xi_{0},\dots,\xi_{k-1},\xi)}{\textsc{vdm}(\xi_{0},\dots,\xi_{k-1})}\geq
M_{k}^{-1}\|P_{k}\|_{K}.$

Hence by definition of $\tau_{k}$, see (18), we have

(25) $\frac{L_{k+1}}{L_{k}}\geq M_{k}^{-1}\|P_{k}\|_{K}\geq
M_{k}^{-1}\tau_{k}^{|\kappa(k)|}.$

By using (23) and iterating the above inequality, we deduce that, for $d\geq
0$

(26) $\displaystyle V_{h_{d}}$ $\displaystyle\geq
L_{h_{d}}=\frac{L_{h_{d}}}{L_{h_{d}-1}}\times\frac{L_{h_{d}-1}}{L_{h_{d}-2}}\times\dots\times\frac{L_{2}}{L_{1}}$
(27)
$\displaystyle\geq\Big{[}\prod_{j=1}^{h_{d}-1}M_{j}\Big{]}^{-1}\Big{[}\prod_{j=1}^{h_{d}-1}\tau_{j}^{|\kappa(j)|}\Big{]}.$

We may re-arrange the terms in the last product above taking into account the
definition of $\mathcal{T}_{i}$ in (20) as follows :

$\prod_{j=1}^{h_{d}-1}\tau_{j}^{[\kappa(j)|}=\prod_{i=1}^{d}\Big{[}\prod_{j=h_{i-1}}^{h_{i}-1}\tau_{j}\Big{]}^{i}=\prod_{i=1}^{d}\mathcal{T}_{i}^{i(h_{i}-h_{i-1})}.$

Now, taking the $l_{d}$-roots in (27), we obtain

(28)
$(V_{h_{d}})^{1/l_{d}}\geq(L_{h_{d}})^{1/l_{d}}\geq\Big{[}\prod_{j=1}^{h_{d}-1}M_{j}\Big{]}^{-1/l_{d}}\Big{[}\prod_{i=1}^{d}\mathcal{T}_{i}^{i(h_{i}-h_{i-1})}\Big{]}^{1/l_{d}}.$

Now, passing to the limit and using Corollary 3.2 and Lemma 2.2, we obtain

$\lim_{d\to\infty}(L_{h_{d}})^{1/l_{d}}=D(K).\qed$

As to prove Relation (15), we proceed as follows. From Inequalities (25) and
(28), we get

$\Big{[}\prod_{j=1}^{h_{d}-1}M_{j}\Big{]}\Big{[}\prod_{i=1}^{d}\mathcal{T}_{i}^{i(h_{i}-h_{i-1})}\Big{]}\leq\prod_{k=0}^{h_{d}-1}\|P_{k}\|_{K}\leq
V_{h_{d}}.$

Then, we take the $l_{d}-$roots and pass to the limit as $d\to\infty$ which
leads to (15).

######  Remark 1.

Due to inequalities (23) and (25) from the above proof, we obtain a stronger
version of Theorem 2.3, namely

(29) $D(K)=\lim_{k\to+\infty}L_{k}^{1/l_{d(k)}},\quad\text{where
}d(k)=|\kappa(k)|.$

##  4\. Intertwining pseudo Leja sequences

We show how to construct a pseudo Leja sequence on
$\mathbb{C}^{p}=\mathbb{C}^{p_{1}}\times\mathbb{C}^{p_{2}}$ out of two pseudo
Leja sequences on $\mathbb{C}^{p_{i}}$, $i=1,2$. To this aim, we use a natural
process introduced by Calvi that generalises a method introduced by Biermann
in the case $p_{i}=1$, see [15]. As indicated in the introduction, in this
section, when necessary, we use a superscript to indicate the ground spaces
for the related object, so that $e_{j}^{(p)}$ denotes the $j$-th monomial on
$\mathbb{C}^{p}$ and so on.

We will use

(30) $\varphi:\mathbb{N}\ni
j\longmapsto\varphi(j)=\big{(}\varphi_{1}(j),\varphi_{2}(j)\big{)}\in\mathbb{N}^{2}$

satisfying, see the introduction for the definition of $\kappa$,

(31)
$\kappa^{(p)}(j)=\left(\kappa^{(p_{1})}\big{(}\varphi_{1}(j)\big{)},\kappa^{(p_{2})}\big{(}\varphi_{2}(j)\big{)}\right),\quad
j\in\mathbb{N}.$

######  Definition 4.1.

Let $A=(a_{j})_{j\in\mathbb{N}}\subset\mathbb{C}^{p_{1}}$ and
$B=(b_{j})_{j\in\mathbb{N}}\subset\mathbb{C}^{p_{2}}$ two sequences of points.
The intertwining sequence of $A$ and $B$ is the sequence
$\Omega=(\omega_{j})_{j\in\mathbb{N}}$ of points in $\mathbb{C}^{p}$,
$p=p_{1}+p_{2}$, denoted by $A\oplus B$ and defined by

(32) $\omega_{j}=\big{(}a_{\varphi_{1}(j)},b_{\varphi_{2}(j)})\big{)},\quad
j\in\mathbb{N}.$

The important point for our purpose is that we can easily express the
vandermondian for an intertwining sequence $\Omega=A\oplus B$ in terms of the
factor sequences $A$ and $B$. This is specified below.

We say that $X=(x_{0},\dots,x_{N-1})$ is completely ordered unisolvent if

(33) $\textsc{vdm}(x_{0},\dots,x_{i-1})\not=0,\quad\text{for all
}i=1,\dots,N.$

Every section $(\xi_{j}:\ j=0,\dots,N-1)$ of a pseudo Leja sequence
$(\xi_{j})_{j\in\mathbb{N}}$ is completely ordered unisolvent (the compact
sets being assumed determining). In view of [15, Theorem 3.3], the notion of
completely ordered unisolvent array is stable by intertwining.

Given a sequence $X=(x_{j})_{j\in\mathbb{N}}$, we set
$X_{j}=(x_{0},\dots,x_{j-1})$. The following result is a consequence of [15,
Proposition 2.5].

######  Proposition 4.1.

Let $X=(x_{0},\dots,x_{N-1})$ be a completely ordered unisolvent array. We
have

(34)
$\textsc{vdm}(x_{0},\dots,x_{N-1})=\prod_{j=0}^{N-1}(e_{j}-L_{X_{j}}e_{j})(x_{j})$

where $L_{X_{j}}$ is the Lagrange interpolation operator at the points
$X_{j}=(x_{0},\dots,x_{j-1})$ with the convention that $L_{X_{0}}f=0$ and
$e_{j}$ is the $j$-th monomial, see (3).

###### Proof.

The proof follows the reasoning used to prove [15, Proposition 2.5]. ∎

The next proposition follows from [15, Theorem 4.2].

######  Proposition 4.2.

Let $A$, $B$ and $\Omega=A\oplus B$ as above. We have

(35)
$\Big{(}e_{j}^{(p)}-L_{\Omega_{j}}e_{j}^{(p)}\Big{)}(z)=\big{(}e_{\varphi_{1}(j)}^{(p_{1})}-L_{A_{\varphi_{1}(j)}}e_{\varphi_{1}(j)}^{(p_{1})}\big{)}(z^{1})\cdot\big{(}e_{\varphi_{2}(j)}^{(p_{2})}-L_{B_{\varphi_{2}(j)}}e_{\varphi_{2}(j)}^{(p_{2})}\big{)}(z^{2})$

for all $z=(z^{1},z^{2})\in\mathbb{C}^{p_{1}}\times\mathbb{C}^{p_{2}}$ and for
all $j\in\mathbb{N}$.

We now combine Proposition 4.1 and Proposition 4.2 to deduce the following
result. Obverse that it involves three forms of vandermondians, with variables
$\omega_{i}$ in $\mathbb{C}^{p}$, $a_{i}$ in $\mathbb{C}^{p_{1}}$, and $b_{i}$
in $\mathbb{C}^{p_{2}}$.

######  Theorem 4.3.

Let $A$, $B$ and $\Omega=A\oplus B$ as above. If, for all
$j\in\mathbb{N}^{\star}$, $A_{j}$ and $B_{j}$ are completely ordered
unisolvent arrays, then for all $j\in\mathbb{N}^{\star}$, we have

(36)
$\textsc{vdm}(\omega_{0},\dots,\omega_{j-1},z)=P_{j}(z)\cdot\textsc{vdm}(\omega_{0},\dots,\omega_{j-1}),$

where, for $z=(z^{1},z^{2})$,

(37) $\displaystyle P_{j}(z)$
$\displaystyle=(e_{j}^{(p)}-L_{\Omega_{j}}e_{j}^{(p)})(z)$ (38)
$\displaystyle=\big{(}e_{\varphi_{1}(j)}^{(p_{1})}-L_{A_{\varphi_{1}(j)}}e_{\varphi_{1}(j)}^{(p_{1})}\big{)}(z^{1})\cdot\big{(}e_{\varphi_{2}(j)}^{(p_{2})}-L_{B_{\varphi_{2}(j)}}e_{\varphi_{2}(j)}^{(p_{2})}\big{)}(z^{2})$

Hence, if $\varphi_{i}(j)>0$ for $i=1,2$,

(39)
$P_{j}(z)=\frac{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1},z^{1})}{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1})}\cdot\frac{\textsc{vdm}(b_{0},\dots,b_{\varphi_{2}(j)-1},z^{2})}{\textsc{vdm}(b_{0},\dots,b_{\varphi_{2}(j)-1})}.$

While if, say, $\varphi_{1}(j)>0$ and $\varphi_{2}(j)=0$,

(40)
$P_{j}(z)=\frac{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1},z^{1})}{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1})}.$

Now, we state the main result of this section.

######  Theorem 4.4.

Let $K_{i}$, $i=1,2$, be a determining compact subset in $\mathbb{C}^{p_{i}}$.
Let $A=(a_{j})_{j\in\mathbb{N}}\subset K_{1}$,
$B=(b_{j})_{j\in\mathbb{N}}\subset K_{2}$ and $\Omega=A\oplus B$. The
following assertions are equivalent:

  1. (1)

$\Omega$ is a pseudo Leja sequence for $K=K_{1}\times K_{2}$.

  2. (2)

$A$ and $B$ are pseudo Leja sequences for $K_{1}$ and $K_{2}$ respectively.

###### Proof.

We prove
$\eqref{assertion:intertwining_pL1}\implies\eqref{assertion:intertwining_pL2}$.
Supposing that $\Omega$ is a pseudo Leja sequence for $K$ of Edrei growth
$(M_{j})_{j\geq 1}$. We prove that $A$ is a pseudo Leja sequence for $K_{1}$.
Let $i\in\mathbb{N}$ and $N\in\mathbb{N}$ such that $\varphi(N)=(i,0)$. From
the definition of a pseudo Leja sequence, we have

$M_{N}|\textsc{vdm}(\omega_{0},\dots,\omega_{N-1},\omega_{N})|\geq\max_{z\in
K}|\textsc{vdm}(\omega_{0},\dots,\omega_{N-1},z)|.$

Applying Theorem 4.3 to both sides of the above inequality, we obtain

$M_{N}|P_{N}(\omega_{N})|\geq\max_{z\in K}|P_{N}(z)|.$

Since $\omega_{N}=(a_{i},b_{0})$, in view of (40), the above inequality
reduces to

$M_{N}|\textsc{vdm}(a_{0},\dots,a_{i-1},a_{i})|\geq\max_{z^{1}\in
K_{1}}|\textsc{vdm}(a_{0},\dots,a_{i-1},z^{1})|.$

Therefore, the sequence $A$ will satisfy the inequality property of a pseudo
Leja sequence on condition that the growth sequence

$M^{\prime}_{i}=M_{N},\quad\varphi(N)=(i,0)$

satisfies the Edrei growth condition in (11), that is

$\lim_{d\to+\infty}\max_{h_{d-1}\leq j<h_{d}}(M^{\prime}_{j})^{1/d}=1,$

where $h_{s}=h_{s}^{(p_{1})}$ is the dimension of the space of polynomial of
degree $s$ on the ground space of $A$, that is $\mathbb{C}^{p_{1}}$. Let
$d\geq 1$ and let

$\mathcal{M}_{i}=M^{\prime}_{i}=\max_{h^{(p_{1})}_{d-1}\leq
j<h^{(p_{1})}_{d}}M^{\prime}_{j}.$

As previously, we consider the number $N=N(i)$ such that $\varphi(N)=(i,0)$.
Therefore, $\kappa^{(p)}(N)=(\kappa^{(p_{1})}(i),0)$ and
$|\kappa^{(p)}(N)|=|\kappa^{(p_{1})}(i)|=d$. It follows that
$N\in\\{h_{d-1}^{(p)},\dots,h_{d}^{(p)}-1\\}$ and hence

$\mathcal{M}_{i}\leq\max_{h^{(p)}_{d-1}\leq j<h^{(p)}_{d}}M_{j}.$

Taking the $1/d$ root and passing to the limit as $d\to\infty$, we obtain the
result, taking into account that $\mathcal{M}_{i}\geq 1$, thanks to the growth
condition of pseudo Leja sequences. Thus, $A$ is a pseudo Leja sequence of
Edrei growth $(M^{\prime}_{i})_{i\geq 1}$. Of course, the same reasoning shows
that $B$ is also a pseudo Leja sequence for $K_{2}$.

We now prove
$\eqref{assertion:intertwining_pL2}\implies\eqref{assertion:intertwining_pL1}$.
We assume that $A$ is a pseudo Leja sequence for $K_{1}$ of Edrei growth
$(M_{j}^{\prime})_{j\geq 1}$ and $B$ is a pseudo Leja sequence for $K_{2}$ of
Edrei growth $(M_{j}^{\prime\prime})_{j\geq 1}$. We show that $\Omega=A\oplus
B$ is a pseudo Leja sequence for $K$.

Fix $j\in\mathbb{N}^{\star}$. If $\varphi_{i}(j)>0$, $i=1,2$, from Theorem
4.3, we have

$|\textsc{vdm}(\omega_{0},\dots,\omega_{j-1},\omega_{j})|=|P_{j}(\omega_{j})|\cdot|\textsc{vdm}(\omega_{0},\dots,\omega_{j-1})|,$

where

$P_{j}(\omega_{j})=\frac{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1},a_{\varphi_{1}(j)})}{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1})}\cdot\frac{\textsc{vdm}(b_{0},\dots,b_{\varphi_{2}(j)-1},b_{\varphi_{2}(j)})}{\textsc{vdm}(b_{0},\dots,b_{\varphi_{2}(j)-1})}.$

In view of (40), one readily check that the above formula remains true when
$\varphi_{i}(j)=0$, $i=1$ or $2$. Using the hypothesis that the factor
sequences are pseudo Leja sequences, we have

$\displaystyle
M^{\prime}_{\varphi_{1}(j)}M^{\prime\prime}_{\varphi_{2}(j)}|P_{j}(\omega_{j})|$
$\displaystyle\geq\max_{z^{1}\in
K_{1}}|\frac{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1},z^{1})}{\textsc{vdm}(a_{0},\dots,a_{\varphi_{1}(j)-1})}|\cdot\max_{z^{2}\in
K_{2}}|\frac{\textsc{vdm}(b_{0},\dots,b_{\varphi_{2}(j)-1},z^{2})}{\textsc{vdm}(b_{0},\dots,b_{\varphi_{2}(j)-1})}|$
$\displaystyle=\max_{z\in K}|P_{j}(z)|.$

Therefore,

$M_{j}|\textsc{vdm}(\omega_{0},\dots,\omega_{j})|\geq\max_{z\in
K}|P_{j}(z)\textsc{vdm}(\omega_{0},\dots,\omega_{j-1})|=\max_{z\in
K}|\textsc{vdm}(\omega_{0},\dots,\omega_{j-1},z)|$

where $M_{j}=M^{\prime}_{\varphi_{1}(j)}M^{\prime\prime}_{\varphi_{2}(j)}$
with the convention that $M^{\prime}_{0}=M^{\prime\prime}_{0}=1$. It remains
to prove that

(41) $\lim_{d\to+\infty}\max_{h^{(p)}_{d-1}\leq j<h^{(p)}_{d}}M_{j}^{1/d}=1.$

Fix $d\geq 1$. Let $j_{d}\in\\{h_{d-1}^{(p)},\dots,h_{d}^{(p)}-1\\}$ such that

$M_{j_{d}}=\max_{h^{(p)}_{d-1}\leq i<h^{(p)}_{d}}M_{j}.$

From the definition of $\varphi$, see (2), we have

(42)
$\displaystyle\kappa^{(p)}(j)=\big{(}\kappa^{(p_{1})}(\varphi_{1}(j)),\kappa^{(p_{2})}(\varphi_{2}(j))\big{)}$
(43) $\displaystyle\implies$ $\displaystyle
d=|\kappa^{(p)}(j)|=|\kappa^{(p_{1})}(\varphi_{1}(j))|+|\kappa^{(p_{2})}(\varphi_{2}(j))|.$

Let $\delta_{i}(d)=|\kappa^{(p_{i})}(\varphi_{i}(j_{d}))|$ for $i=1,2$. Now we
have

$M_{j_{d}}^{1/d}=\left((M^{\prime}_{\varphi_{1}(j_{d})})^{1/|\delta_{1}(d)|}\right)^{|\delta_{1}(d)|/d}\times\left((M^{\prime\prime}_{\varphi_{2}(j_{d})})^{1/|\delta_{2}(d)|}\right)^{|\delta_{2}(d)|/d}.$

We consider two cases. If say $\delta_{1}(d)$ is bounded then, in view of
(43), $\delta_{2}(d)/d\to 1$, and (41) follows from the fact that
$\delta_{1}(d)/d\to 0$ and that, $B$ being a pseudo Leja sequence,
$M^{\prime\prime}$ satisfies the required growth condition. Since, by (43),
both $\delta_{i}(d)/d$ cannot be bounded, it remains only to the study the
case for which they are unbounded hence, since they are increasing, tending to
$\infty$. In that case, (41) follows from the fact that $M^{\prime}$ and
$M^{\prime\prime}$ satisfies the Edrei growth condition and that
$\delta_{i}(d)/d$ is bounded by $1$, $i=1,2$. ∎

Theorem 4.4 is a generalisation of Irigoyen’s recent result in [19, Theorem 1
] about the intertwining of Leja sequences when $p_{1}=p_{2}=1$. It is well
known that the intertwining of two good univariate sequences of interpolation
points (good in the sense given in the introduction) is also good for the
interpolation of holomorphic functions in $\mathbb{C}^{2}$. This property was
first proved by Siciak in [23] but can also be found in [4], [5] where
different proofs are proposed.

##  5\. Computing pseudo-Leja points

In practice, we compute finitely many points, even the first few points, see
below, of a pseudo-Leja sequences. Such a computation is based on the
following observation which shows how to obtain pseudo-Leja points on $K$ as
Leja points for a finite subset of $K$.

######  Proposition 5.1.

Let $A$ a finite subset of $K\in\mathbb{C}^{p}$ that is determining for the
space of polynomial of degree $\leq d$ (hence $\textbf{Card}(A)\geq
h_{d}=h_{d}^{(p)}$). We define inductively $h_{d}$ points as follows.

  1. (1)

We choose arbitrarily $\xi_{0}\in A$ and,

  2. (2)

for each $d\in\mathbb{N}$, we compute recursively the points
$\xi_{h_{d-1}},\dots,\xi_{h_{d}-2}$ and $\xi_{h_{d}-1}$ from $A$ such that

(44) $|\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},\xi_{N})|=\max_{\xi\in
A}|\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},\xi)|,$

for $h_{d-1}\leq N<h_{d}$.

Then the points $\xi_{j}$ forms the first $h_{d}$ points of a pseudo-Leja
sequences for $K$ of (constant) Edrei growth $M_{A}$ where

$M_{A}=\max\\{1/\|P\|_{A},\quad\|P\|_{K}=1,\deg(P)\leq d\\}.$

These points are called discrete Leja points for $A$.

###### Proof.

Observe that the finiteness of $M_{A}$ follows from the facts that $A$ is
determining for the space of polynomial of degree $\leq d$ and that the set of
polynomials of bounded degree $d$ and sup norm not bigger than $1$ on $K$ is
compact. This being said, the claim follows immediately from the observation
that for every $h_{d-1}\leq N<h_{d}$, the function

$\xi\longmapsto|\textsc{vdm}(\xi_{1},\dots,\xi_{N-1},\xi)|$

is a polynomial of $n$ complex variables and of degree at most $d$ so that, by
definition of $M_{A}$,

$\max_{\xi\in K}|\textsc{vdm}(\xi_{1},\dots,\xi_{N-1},\xi)|\leq
M_{A}\max_{\xi\in A}|\textsc{vdm}(\xi_{1},\dots,\xi_{N-1},\xi).\qed$

In practice, the smallest the constant $M_{A}$ the better the points. In
general, it is not easy to select a mesh $A$ with low constant $M_{A}$ and low
cardinality. The best algorithm currently available to compute discrete Leja
points is provided in [11]. In short, it is as follows. If we write
$A=\\{a_{1},\dots,a_{M}\\}$ ($M>h_{d}$) and if
$\textbf{P}=\\{P_{1},\dots,P_{h_{d}}\\}$ is an ordered polynomial basis then
applying the standard LU factorisation with row pivoting to the following
rectangular Vandermonde matrix

(45) $V(a_{1},\dots,a_{M};\textbf{P})=[P_{j}(a_{i})]_{1\leq i\leq M,1\leq
j\leq h_{d}}.$

produces a permutation vector $\sigma$ whose first $h_{d}$ components are the
indices of the first $h_{d}$ Leja points in A. During this process, the matrix
$V(a_{1},\dots,a_{M};\textbf{P})$ is progressively modified by operations on
the rows and the pivot row is always selected from the candidates with the
largest absolute value in the first column of the submatrix of interest.
Observe, due to the large dimension of the determinant involved, one can
currently compute points only for low degrees, say $d=10$ for $p=2$.

Finally, we mention a (theoretical) method, based on the same idea as the
above proposition, using the concept of (weakly) admissible mesh (for $K$)
introduced in [16] which consists of a sequence of sets
$(A_{d})_{d}\in\mathbb{N}$, such that $A_{d}\subset K$ is determining for the
space of polynomials of degree $\leq d$ and both $M_{A_{d}}$ and the
cardinality of $A_{d}$ grows subexponentially as $d\to\infty$. The proof is
immediate and is omitted.

######  Proposition 5.2.

Let $K$ a determining compact set in $\mathbb{C}^{p}$. and
$(A_{d})_{d\in\mathbb{N}}$ an admissible mesh for $K$. A pseuso-Leja sequence
$(\xi_{N})_{N\in\mathbb{N}}$ for $K$ of Edrei growth $\overline{M}_{N}$ with

$\overline{M}_{N}=M_{A_{d}},\quad h_{d-1}\leq N<h_{d},$

is obtained as follows :

  1. (1)

$\xi_{0}\in A_{0}$ and,

  2. (2)

for each $d\in\mathbb{N}$, we compute recursively the points
$\xi_{h_{d-1}},\dots,\xi_{h_{d}-2}$ and $\xi_{h_{d}-1}$ from $\mathcal{A}_{d}$
such that

(46) $|\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},\xi_{N})|=\max_{\xi\in
A_{d}}|\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},\xi)|.$

### Acknowledgement

I would like to thank Professor Leokadia Białas-Cież, my research supervisor,
for her patient guidance, enthusiastic encouragement and useful critiques of
this work. I would also like to express my sincere gratitude to Professor
Jean-Paul Calvi for his unconditional help in structuring and writing this
paper.
Funding: This work was partially supported by the National Science Center,
Poland, grant Preludium Bis 1 N. 2019/35/O/ST1/02245.

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Moments Match between the KPZ Equation and the Airy Point Process

Moments Match between the KPZ Equation
and the Airy Point Process\star\star$\star$This paper is a contribution to the
Special Issue on Asymptotics and Universality in Random Matrices, Random
Growth Processes, Integrable Systems and Statistical Physics in honor of Percy
Deift and Craig Tracy. The full collection is available at
http://www.emis.de/journals/SIGMA/Deift-Tracy.html

Alexei BORODIN †‡ and Vadim GORIN †‡

A. Borodin and V. Gorin

† Department of Mathematics, Massachusetts Institute of Technology, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Institute for Information Transmission Problems of Russian Academy of
Sciences, Russia

Received August 09, 2016, in final form October 21, 2016; Published online
October 26, 2016

The results of Amir–Corwin–Quastel, Calabrese–Le Doussal–Rosso, Dotsenko, and
Sasamoto–Spohn imply that the one-point distribution of the solution of the
KPZ equation with the narrow wedge initial condition coincides with that for a
multiplicative statistics of the Airy determinantal random point process.
Taking Taylor coefficients of the two sides yields moment identities. We
provide a simple direct proof of those via a combinatorial match of their
multivariate integral representations.

KPZ equation; Airy point process

60B20; 60H15; 33C10

##  1 Introduction

Since Tracy and Widom’s discovery of the ASEP solvability eight years ago [18,
19, 20], the relationship between the “determinantal” and “non-determinantal”
solvable models in the $(1+1)$-dimensional KPZ (Kardar–Parisi–Zhang)
universality class has largely remained a mystery. One step towards solving
this mystery is the celebrated result of Amir–Corwin–Quastel [1], Calabrese–Le
Doussal–Rosso [8], Dotsenko [10], and Sasamoto–Spohn [17], that provides an
explicit expression for the distribution (or its Laplace transform) of one-
point value of the solution of the KPZ equation with the so-called narrow
wedge initial condition. It can be re-interpreted as saying that this Laplace
transform coincides with the average of a multiplicative statistics of the
Airy determinantal random point process. Although this restatement seems to be
known to experts, we couldn’t find it in this form in the literature, so we
give an exact formulation as Theorem 2.1 below. Such a result is very useful
as it immediately implies that this solution of the KPZ equation
asymptotically at large times has the GUE Tracy–Widom distribution, which is a
display of the KPZ universality, cf. Corwin’s survey [9].

Finding other facts of similar nature has been a challenge so far.

Imamura and Sasamoto [12] proved a similar statement for the O’Connell–Yor
semi-discrete Brownian directed polymer. Unfortunately, the associated
determinantal point process was not governed by a positive measure. Still,
taking the edge limit of this process, they were able to recover Theorem 2.1.
Another representation of the Laplace transform of the O’Connell–Yor partition
function as the average of a multiplicative functional over a signed
determinantal point process can be found in [14].

Very recently, one of the authors found in [4] an identity that relates a
single point height distribution of the (higher spin inhomogeneous) stochastic
six vertex model in a quadrant on one side, and multiplicative statistics of
the Macdonald measures on the other. The ASEP limit of this identity was
worked out in [7]. Taking the KPZ limit of both leads to Theorem 2.1 again.

The goal of this note is to look at Theorem 2.1 from the point of view of
moments, rather than the corresponding distributions. One can study both the
KPZ equation and the Airy point process via their exponential moments. Those
are computationally tractable but they are of limited mathematical use because
the corresponding moment problems are indeterminate. Still, on the KPZ side
physicists were able to consistently use the moments to access the
distributions via the (non-rigorous) replica trick, see [10] and [8] for early
examples.

Our Theorem 2.2 proves the moments identity that corresponds to Theorem 2.1.
The argument is a combinatorial match between known multivariate integral
representations of the moments on both sides. Interestingly, these integral
representations were known long before [1, 8, 10, 17], but their similarity
had not been exploited. We are hoping that the moments point of view will be
beneficial for finding other similar correspondence.

We did attempt to extend the moments correspondence to a two-point identity,
as integral representations on both side are again known. Unfortunately, we
have not been successful in that so far.

##  2 The one-point equality

Let $\mathfrak{a}_{1}\geq\mathfrak{a}_{2}\geq\mathfrak{a}_{3}\geq\cdots$ be
points of the Airy point process111It should not be confused with Airy2
process; the latter is a random continuous curve rather than a random point
process. at $\beta=2$ (see, e.g., [2, 11]) which is a determinantal point
process on $\mathbb{R}$ with correlation kernel

$\displaystyle K_{\rm
Airy}(x,y)=\frac{\operatorname{Ai}(x)\operatorname{Ai}^{\prime}(y)-\operatorname{Ai}^{\prime}(x)\operatorname{Ai}(y)}{x-y}=\int_{0}^{\infty}\operatorname{Ai}(x+a)\operatorname{Ai}(y+a)da.$

Here $\operatorname{Ai}(x)$ is the Airy function.

From the opposite direction, let $\mathcal{Z}(T,X)$ denote the solution of the
stochastic heat equation (see, e.g., [9, 16])

$\displaystyle\frac{\partial}{\partial
T}\mathcal{Z}=\frac{1}{2}\frac{\partial^{2}}{\partial
X^{2}}\mathcal{Z}-\mathcal{Z}\dot{\mathcal{W}},\qquad\mathcal{Z}(0,X)=\delta(X=0),$

where $\dot{\mathcal{W}}$ is the space–time white noise.
$H:=-\log(\mathcal{Z})$ is the Hopf–Cole solution of the Kardar–Parisi–Zhang
stochastic partial differential equation with the narrow-edge initial data.

The following statement is a reformulation of results of [1, 8, 10, 17].

######  Theorem 2.1.

Set $\frac{T}{2}=C^{3}$. Then for each real $C>0$, $u\geq 0$ we have

$\displaystyle\mathbb{E}_{\mathrm{Airy}}\left[\prod_{k=1}^{\infty}\frac{1}{1+u\exp\left(C\mathfrak{a}_{k}\right)}\right]=\mathbb{E}_{\mathrm{KPZ}}\left[\exp\left({-}u\mathcal{Z}(T,0)\exp\left(\frac{T}{24}\right)\right)\right].$
(1)

On the other hand, we could not find the following statement in the existing
literature.

######  Theorem 2.2.

Set $\frac{T}{2}=C^{3}$, and let
$h_{k}(x_{1},x_{2},\dots)=\sum\limits_{i_{1}\leq i_{2}\leq\dots\leq
i_{k}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}$ be the complete symmetric
homogeneous function in variables $x_{1},x_{2},\dots$. Then for each $C>0$,
$k=1,2,\dots$ we have

$\displaystyle\mathbb{E}_{\mathrm{Airy}}\left[h_{k}\left(\exp\left(C\mathfrak{a}_{1}\right),\exp\left(C\mathfrak{a}_{2}\right),\dots\right)\right]=\mathbb{E}_{\mathrm{KPZ}}\left[\frac{\mathcal{Z}(T,0)^{k}}{k!}\right]\exp\left(k\frac{T}{24}\right).$
(2)

Our proof of Theorem 2.2 is based on direct comparison of contour integral
formulas: for the right-hand side of (2) such formula is known as a solution
for the attractive delta Bose gas equation, cf. the discussion in [6, Section
6.2], while for the left-hand side it can be computed through the Laplace
transform of the correlation kernel $K_{\rm Airy}$.

######  Remark 2.3.

Expanding formally the result of Theorem 2.1 into power series in $u$ and
evaluating the coefficients, one gets the result of Theorem 2.2 and vice
versa. However, these theorems are not equivalent: The $u$ power series
expansion of the left-hand side of (2.1) fails to converge for any $u\neq 0$.
Below we provide two different proofs for Theorems 2.1 and 2.2, respectively.

The following corollary is present in [1, 8, 10, 17], but it seems reasonable
for us to give a proof using Theorem 2.1 above only, without appealing to the
explicit evaluation of either side.

######  Corollary 2.4.

The following convergence in distribution holds:

$\displaystyle\lim_{T\to+\infty}\left[\left(\frac{2}{T}\right)^{1/3}\left(\ln(Z(T,0))+\frac{T}{24}\right)\right]=\mathfrak{a}_{1}.$

###### Proof.

Take $a\in\mathbb{R}$ and set $u=\exp\bigl{(}{-}(T/2)^{1/3}a\bigr{)}$. Then
the left-hand side of (1) is

$\displaystyle\mathbb{E}\left[\prod_{k=1}^{\infty}\frac{1}{1+\exp\bigl{(}(T/2)^{1/3}(\mathfrak{a}_{k}-a)\bigr{)}}\right].$
(3)

We claim that the random variable under expectation in (3) almost surely
converges as $T\to\infty$ to the indicator function of the event
$\mathfrak{a}_{1}<a$. Indeed, if $\mathfrak{a}_{1}>a$, then the expression in
(3) converges to $0$ as $T\to\infty$. On the other hand, if
$\mathfrak{a}_{1}<a$, then (taking into the account that
$\sum\limits_{k=1}^{\infty}\exp(\mathfrak{a}_{k})$ is almost surely finite, as
follows from the finiteness of its expectation, which is explicitly known,
see, e.g., [15, Section 2.6.1]) the same expression converges to $1$. Since
the random variable under expectation is almost surely between $0$ and $1$,
the almost sure convergence implies convergence of expectations, and therefore

$\displaystyle\lim_{T\to+\infty}\mathbb{E}\left[\prod_{k=1}^{\infty}\frac{1}{1+\exp\bigl{(}(T/2)^{1/3}(\mathfrak{a}_{k}-a)\bigr{)}}\right]={\rm
Prob}(\mathfrak{a}_{1}<a).$ (4)

On the other hand, the right side of (1) is

$\displaystyle\mathbb{E}\left[\exp\left(-\exp\left((T/2)^{1/3}\left(\frac{\ln(\mathcal{Z}(T,0))+T/24}{(T/2)^{1/3}}-a\right)\right)\right)\right].$
(5)

Observe that for any random variable $\xi$, the expression
$\exp\bigl{(}{-}\exp((T/2)^{1/3}(\xi-a))\bigr{)}$ is almost surely between $0$
and $1$, converges to $0$ as $T\to+\infty$ if $\xi>a$, and converges to $1$ if
$\xi<a$. Therefore, using the fact that $\mathfrak{a}_{1}$ has a continuous
distribution, (5) as $T\to\infty$ behaves as (see, e.g., [6, Lemma 4.1.39] for
more details)

$\displaystyle{\rm
Prob}\left(\frac{\ln(\mathcal{Z}(T,0))+T/24}{(T/2)^{1/3}}<a\right)+o(1).$ (6)

Equating (4) to (6) we are done. ∎

##  3 Proofs of Theorems 2.1 and 2.2

###### Proof of Theorem 2.1.

The expectation $\mathbb{E}\exp(-u\mathcal{Z}(T,0)$ admits a formula as a
Fredholhm determinant, which was discovered in [1, 8, 10, 17]. Following [5,
Section 2.2.1], this formula reads

$\displaystyle\mathbb{E}\big{[}\exp\bigl{(}{-}u\mathcal{Z}(T,0)\exp(T/24)\bigr{)}\big{]}=1+\sum_{L=1}^{\infty}\frac{(-1)^{L}}{L!}\int\limits_{0}^{\infty}dx_{1}\cdots\int\limits_{0}^{\infty}dx_{L}\det\left[K_{u}(x_{i},x_{j})\right]^{L}_{i,j=1},$
(7)

where

$\displaystyle
K_{u}(x,x^{\prime})=\int_{-\infty}^{\infty}\frac{dr}{1+\frac{1}{u}\exp\big{(}(T/2)^{1/3}r\big{)}}\operatorname{Ai}(x-r)\operatorname{Ai}(x^{\prime}-r).$

On the other hand, the left-hand side of (1) is a multiplicative function of a
determinantal point process and, therefore, also admits a Fredholm determinant
formula, see, e.g., [3, equation (2.4)]:

$\displaystyle\mathbb{E}\left[\prod_{k=1}^{\infty}\frac{1}{1+u\exp\left(C\mathfrak{a}_{k}\right)}\right]=\det\left[1-\left(1-\frac{1}{1+u\exp(Cr)}\right)K_{\rm
Airy}(r,r^{\prime})\right]_{L_{2}(\mathbb{R})}$
$\displaystyle=1+\sum_{L=1}^{\infty}\frac{(-1)^{L}}{L!}\int_{-\infty}^{\infty}dy_{1}\cdots\int_{-\infty}^{\infty}dy_{L}\left(\prod_{k=1}^{L}\frac{1}{1+\frac{1}{u}\exp(-Cy_{k})}\right)\det\left[K_{\rm
Airy}(y_{i},y_{j})\right]_{i,j=1}^{L}.\\!\\!\\!$ (8)

One immediately sees that upon the change of variables $r_{i}=-y_{i}$ and
identification $\frac{T}{2}=C^{3}$, the formulas (7) and (8) are the same. ∎

###### Proof of Theorem 2.2.

The moments $\mathbb{E}\mathcal{Z}(T,0)^{k}$ are known through solving the
attractive delta Bose gas equation, cf. the discussion in [6, Section 6.2].
Following [5, Lemma 4.1], we have

$\displaystyle\mathbb{E}\big{[}\mathcal{Z}(T,0)^{k}\big{]}=\int_{a_{1}-{\rm
i}\infty}^{a_{1}+{\rm i}\infty}\frac{dz_{1}}{2\pi{\rm
i}}\cdots\int_{a_{k}-{\rm i}\infty}^{a_{k}+{\rm
i}\infty}\frac{dz_{k}}{2\pi{\rm i}}\prod_{1\leq A<B\leq
k}\frac{z_{A}-z_{B}}{z_{A}-z_{B}-1}\cdot\prod_{j=1}^{k}\exp\left(\frac{T}{2}z_{j}^{2}\right),$
(9)

where the real numbers $a_{1},\dots,a_{k}$ satisfy $a_{1}\gg a_{2}\gg\cdots\gg
a_{k}$. It is convenient for us to modify the contours of integration in (9)
to the imaginary axis ${\rm i}\mathbb{R}$. One collects certain residues in
such a deformation, and the final result is read from [5, equation (13)] to be

$\displaystyle\mathbb{E}\left[\frac{\mathcal{Z}(T,0)^{k}}{k!}\right]=\sum_{\begin{smallmatrix}\lambda\vdash
k\\\
\lambda=1^{m_{1}}2^{m_{2}}\cdots\end{smallmatrix}}\frac{1}{m_{1}!m_{2}!\cdots}\int_{-{\rm
i}\infty}^{{\rm i}\infty}\frac{dw_{1}}{2\pi{\rm i}}\cdots\int_{-{\rm
i}\infty}^{{\rm i}\infty}\frac{dw_{\ell(\lambda)}}{2\pi{\rm
i}}\det\left[\frac{1}{w_{j}+\lambda_{j}-w_{i}}\right]_{i,j=1}^{\ell(\lambda)}$
$\displaystyle\hphantom{\mathbb{E}\left[\frac{\mathcal{Z}(T,0)^{k}}{k!}\right]=}{}\times\prod_{j=1}^{\ell(\lambda)}\exp\left(\frac{T}{2}\big{(}w_{j}^{2}+(w_{j}+1)^{2}+\cdots+(w_{j}+\lambda_{j}-1)^{2}\big{)}\right),$
(10)

where $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots)$ is a partition of $k$
and $\ell(\lambda)$ is the number of non-zero parts $\lambda_{j}$.

Let us now produce a similar expression for the left-hand side of (2). Define
the Laplace transform of the correlation functions of the Airy point process
through

$\displaystyle R(c_{1},\dots,c_{n})=\int_{\mathbb{R}^{n}}e^{(c\cdot
x)}\det[K_{\rm Airy}(x_{i},x_{j})]_{i,j=1}^{n}dx_{1}\cdots dx_{n},\qquad
c_{1},c_{2},\dots,c_{n}>0.$

The definition of the Airy point process implies that for a partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\dots\geq\lambda_{\ell})=1^{m_{1}}2^{m_{2}}\cdots$
one has

$\displaystyle\mathbb{E}\bigl{[}m_{\lambda}(\exp(C\mathfrak{a}_{1}),\exp(C\mathfrak{a}_{2}),\dots)\bigr{]}=\frac{1}{m_{1}!m_{2}!\cdots}R(C\lambda_{1},\dots,C\lambda_{\ell}),$

where $m_{\lambda}(y_{1},y_{2},\dots)$ is the monomial symmetric function in
variables $y_{1},y_{2},\dots$, as in [13, Chapter I]. Expanding $h_{k}$ into
linear combination of $m_{\lambda}$’s, we can then write

$\displaystyle\mathbb{E}\bigl{[}h_{k}(\exp(C\mathfrak{a}_{1}),\exp(C\mathfrak{a}_{2}),\dots)\bigr{]}=\sum_{\begin{smallmatrix}\lambda\vdash
k\\\
\lambda=1^{m_{1}}2^{m_{2}}\cdots\end{smallmatrix}}\frac{1}{m_{1}!m_{2}!\cdots}R\bigl{(}C\lambda_{1},\dots,C\lambda_{\ell(\lambda)}\bigr{)},$
(11)

where the summation goes over all partitions of $k$. Comparing (11) with (10),
we see that it remains to identify the contour integrals over imaginary axis
in (10) with $R\bigl{(}C\lambda_{1},\dots,C\lambda_{\ell(\lambda)}\bigr{)}$.

The rest is based on the following identity that can be found in [15, Lemma
2.6]:

$\displaystyle\int_{-\infty}^{+\infty}e^{xz}\operatorname{Ai}(z+a)\operatorname{Ai}(z+b)dz=\frac{1}{2\sqrt{\pi
x}}\exp\left(\frac{x^{3}}{12}-\frac{a+b}{2}x-\frac{(a-b)^{2}}{4x}\right),\qquad
x>0.$

Its immediate corollary is (we use an agreement $z_{n+1}=z_{1}$ and
$s_{n+1}=s_{1}$ here, and also assume $c_{1},\dots,c_{n}>0$)

$\displaystyle{\mathcal{E}}(c_{1},\dots,c_{n}):=\int_{\mathbb{R}^{n}}e^{(c\cdot
z)}\prod_{i=1}^{n}K_{\rm
Airy}(z_{i},z_{i+1})dz=\frac{1}{2^{n}\pi^{n/2}}\frac{e^{\sum
c_{i}^{3}/12}}{\prod\limits_{i=1}^{n}\sqrt{c_{i}}}$
$\displaystyle\hphantom{{\mathcal{E}}(c_{1},\dots,c_{n}):=}{}\times\int_{s_{1}\geq
0}\cdots\int_{s_{n}\geq
0}\exp\left(-\sum_{i=1}^{n}\frac{(s_{i}-s_{i+1})^{2}}{4c_{i}}-\sum_{i=1}^{n}\frac{s_{i}+s_{i+1}}{2}c_{i}\right)\prod_{i=1}^{n}ds_{i}.$

Using the Gaussian integrals in variables $z_{1},\dots,z_{n}$, the last
formula is converted into

$\displaystyle{\mathcal{E}}(c_{1},\dots,c_{n})=\frac{1}{(2\pi)^{n}}e^{\sum
c_{i}^{3}/12}\int_{s_{1}\geq 0}ds_{1}\cdots\int_{s_{n}\geq
0}ds_{n}\int_{z_{1}\in\mathbb{R}}dz_{1}\cdots\int_{z_{n}\in\mathbb{R}}dz_{n}$
$\displaystyle\hphantom{{\mathcal{E}}(c_{1},\dots,c_{n})=}{}\times\exp\left(\sum_{i=1}^{n}\big{(}{-}c_{i}z_{i}^{2}+{\rm
i}(z_{i}-z_{i+1})s_{i}-(c_{i}+c_{i+1})s_{i}/2\big{)}\right).$ (12)

Since ${\rm i}(z_{i}-z_{i+1})$ has zero real part, we can integrate over
$s_{i}$ in (12), arriving at the formula:

$\displaystyle{\mathcal{E}}(c_{1},\dots,c_{n})=\frac{e^{\sum
x_{i}^{3}/12}}{(2\pi)^{n}}\int_{z_{1}\in\mathbb{R}}\cdots\int_{z_{n}\in\mathbb{R}}\exp\left(-\sum_{i=1}^{n}c_{i}z_{i}^{2}\right)\prod_{i=1}^{n}\frac{dz_{i}}{-{\rm
i}(z_{i}-z_{i+1})+\frac{c_{i}+c_{i+1}}{2}}.$ (13)

We can now write the formula for $R(c_{1},\dots,c_{n})$ (we subdivide a
permutation into cycles, use (13) and then combine back):

$\displaystyle R(c_{1},\dots,c_{n})=\int_{\mathbb{R}^{n}}dx_{1}\cdots
dx_{n}\sum_{\sigma\in{\mathfrak{S}}(n)}(-1)^{\sigma}\prod_{j=1}^{n}e^{x_{j}c_{j}}K_{\rm
Airy}(x_{j},x_{\sigma(j)})$ $\displaystyle=\frac{e^{\sum
c_{i}^{3}/12}}{(2\pi)^{n}}\int_{z_{1}\in\mathbb{R}}\cdots\int_{z_{n}\in\mathbb{R}}\exp\left(-\sum_{i=1}^{n}c_{i}z_{i}^{2}\right)\sum_{\sigma\in{\mathfrak{S}}(n)}(-1)^{\sigma}\prod_{i=1}^{n}\frac{dz_{i}}{-{\rm
i}(z_{i}-z_{\sigma(i)})+\frac{c_{i}+c_{\sigma(i)}}{2}}$
$\displaystyle=\frac{e^{\sum
c_{i}^{3}/12}}{(2\pi)^{n}}\int_{z_{1}\in\mathbb{R}}dz_{1}\cdots\int_{z_{n}\in\mathbb{R}}dz_{n}\exp\left(-\sum_{i=1}^{n}c_{i}z_{i}^{2}\right)\det\left[\frac{1}{\big{(}{-}{\rm
i}z_{i}+\frac{c_{i}}{2}\big{)}+\big{(}{\rm
i}z_{j}+\frac{c_{j}}{2})}\right]_{i,j=1}^{n}.$

######  Remark 3.1.

We can use the Cauchy determinant formula

$\displaystyle\det\left[\frac{1}{a_{i}+b_{j}}\right]_{i,j=1}^{n}=\prod_{i=1}^{n}\frac{1}{a_{i}+b_{i}}\prod_{1\leq
i<j\leq n}\frac{(a_{i}-a_{j})(b_{i}-b_{j})}{(a_{i}+b_{j})(a_{j}+b_{i})}$

with $a_{i}=-{\rm i}z_{i}+c_{i}/2$, $b_{i}={\rm i}z_{i}+c_{i}/2$ to simplify
the last determinant.

We now take a partition $\lambda\vdash k$ with $\ell(\lambda)=n$, set
$c_{i}=C\lambda_{i}$ and make a change of variables

$\displaystyle{\rm i}z_{j}=Cw_{j}+\frac{C\lambda_{j}}{2}-\frac{C}{2}$

to get (note that we deformed the contours to the imaginary axis; we do not
pick up any residues in such a deformation)

$\displaystyle
R(C\lambda_{1},\dots,C\lambda_{n})=\frac{\exp\Big{(}C^{3}\sum\limits_{i=1}^{n}\lambda_{i}^{3}/12\Big{)}}{(2\pi{\rm
i})^{n}}\int_{-{\rm i}\infty}^{{\rm i}\infty}dw_{1}\cdots\int_{-{\rm
i}\infty}^{{\rm i}\infty}dw_{n}$
$\displaystyle\hphantom{R(C\lambda_{1},\dots,C\lambda_{n})=}{}\times\exp\left(C^{3}\sum_{i=1}^{n}\lambda_{i}(w_{i}+\lambda_{i}/2-1/2)^{2}\right)\det\left[\frac{1}{w_{j}+\lambda_{j}-w_{i}}\right]_{i,j=1}^{n}.$
(14)

It remains to simplify the exponents:

$\displaystyle\exp\left[C^{3}\sum_{i=1}^{n}\left(\frac{\lambda_{i}^{3}}{12}+\lambda_{i}\left(w_{i}+\frac{\lambda_{i}}{2}-\frac{1}{2}\right)^{2}\right)\right]$
$\displaystyle\qquad{}=\exp\left[C^{3}\sum_{i=1}^{n}\left(\lambda_{i}w_{i}^{2}+\frac{\lambda_{i}^{3}}{3}+\frac{\lambda_{i}}{4}+w_{i}\lambda_{i}^{2}-w_{i}\lambda_{i}-\frac{\lambda_{i}^{2}}{2}\right)\right]$
$\displaystyle\qquad{}=\exp\left[C^{3}\sum_{i=1}^{n}\left(\lambda_{i}w_{i}^{2}+\lambda_{i}(\lambda_{i}-1)w_{i}+\frac{\lambda_{i}(\lambda_{i}-1)(2\lambda_{i}-1)}{6}-\frac{\lambda_{i}}{12}\right)\right]$
$\displaystyle\qquad{}=\exp\left[C^{3}\sum_{i=1}^{n}\big{(}w_{i}^{2}+(w_{i}+1)^{2}+\dots+(w_{i}+\lambda_{i}-1)^{2}\big{)}-C^{3}\frac{k}{12}\right].$
(15)

Combining (11) with (14), (15) and identifying $C^{3}=\frac{T}{2}$. we arrive
at (10) multiplied by $\exp(-kT/24)$. ∎

### Acknowledgements

A.B. was partially supported by the NSF grants DMS-1056390 and DMS-1607901.
V.G. was partially supported by the NSF grant DMS-1407562 and by the Sloan
Research Fellowship.

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