# Peaks of cylindric plane partitions

Dan Betea1<EMAIL_ADDRESS>supported by FWO Flanders project EOS
30889451. Alessandra Occelli2,3<EMAIL_ADDRESS>supported by the HyLEF ERC starting grant 2016. This project was completed
while A.O. was a postdoctoral fellow at MSRI during the Program “Universality
and Integrability in Random Matrix Theory and Interacting Particle Systems”. 1
Department of Mathematics, KU Leuven, Belgium
2 Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal
3 Mathematical Sciences Research Institute, Berkeley, CA, USA

###### Abstract

We study the asymptotic distribution, as the volume parameter goes to 1, of
the peak (largest part) of finite- or slowly-growing-width cylindric plane
partitions weighted by their trace, seam, and volume. There are two natural
asymptotic regimes depending on the trace/seam parameters, and in both cases
we obtain asymptotics governed by finite temperature (periodic) analogues of
the Bessel and Airy gap probabilities from random matrix theory. In
particular, the distributions we obtain interpolate _in more than one way_
between two well-known extremal value distributions: the Gumbel distribution
of maxima of iid random variables and the Tracy–Widom distribution of maxima
of eigenvalues of random Hermitian matrices. We also interpret our results in
terms of last passage percolation on a cylinder, which yields to interesting
connections to the Kardar–Parisi–Zhang equation.

##  1 Introduction and main results

### Background and motivation.

Cylindric plane partitions are better explained in a picture than a formula:
they are plane partitions wrapped around the cylinder as in Figure 1 (left).
They have been studied in the combinatorial and probabilistic literature in
various guises, and we mention the works of Gessel–Krattenthaler [9] and
Borodin [6] as two notable examples. We note that the latter article is
probabilistic in nature and deals with the local nature of the correlation
functions “in the bulk” as the size of the partitions (or the volume
parameter) grows (resp. approaches 1).

It has been known for a while that cylindric boundary conditions for
combinatorial objects correspond to finite temperature systems in the sense of
quantum mechanics, with inverse temperature the same as the cylinder’s
circumference. An example to illustrate both the combinatorics and the
mathematical physics is the cylindric Plancherel measure discussed in [3] (and
introduced in [6]). The motivation for this paper stems from there: by
studying simple distributions on cylindric plane partitions, we obtain
interesting asymptotic behavior for their peaks that “crosses over” between a
large (infinite) temperature regime governed by the Gumbel distribution
(universally the asymptotic maximum of iid random variables that have
Gaussian-like tails) and a small (zero) temperature regime governed by the
Tracy–Widom [22] distribution (universally the asymptotic maximum of
correlated spectra of Hermitian matrices with iid entries).

### Main result.

In this work we initiate the study of the peaks/largest parts of such
partitions. We limit ourselves to a cylinder of finite or slowly growing
circumference $2N$ and consider the case of volume-trace-and-seam-distributed
such partitions that contain only one peak (and also only one “bottom”). In
technical terms, we look at cylindric plane partitions $\Lambda$ that are in
bijection with a sequence of $2N$ interlacing ordinary integer partitions as
follows:

$\Lambda=\\{\mu=\lambda^{(-N)}\prec\cdots\prec\lambda^{(-1)}\prec\lambda^{(0)}=\lambda\succ\lambda^{(1)}\succ\cdots\succ\lambda^{(N)}=\mu\\}.$
(1)

The largest (ordinary) partition $\lambda$ gives the peak $\lambda_{1}$ of
$\Lambda$ (the size of the inner-most central column of boxes), and also the
trace: $\mathrm{tr}\,\Lambda:=|\lambda|$ where $|\lambda|=\sum_{i}\lambda_{i}$
denotes the size of an integer partition. The smallest partition $\mu$ gives
the seam of $\Lambda$: $\mathrm{sm}\,\Lambda:=|\mu|$. The volume of $\Lambda$,
denoted $|\Lambda|$, is the number of cubes in it: $|\Lambda|:=\sum_{-N\leq
i<N-1}|\lambda^{(i)}|$.

We fix a positive integer $N$ and two numbers $0\leq a,q\leq 1$ such that
$aq<1$. The probability distribution considered throughout is the following
trace-seam-volume one:

$\mathbb{P}(\Lambda)=\frac{a^{\mathrm{tr}\,\Lambda}\cdot(a^{-1}q^{-N})^{\mathrm{sm}\,\Lambda}\cdot
q^{|\Lambda|}}{Z}=\frac{a^{|\lambda|}\cdot(a^{-1}q^{-N})^{|\mu|}\cdot
q^{\sum_{i}|\lambda^{(i)}|}}{Z}$ (2)

where $Z=(q^{-N};q^{-N})^{-1}_{\infty}\prod_{1\leq i,j\leq
N}(q^{i+j-1};q^{N})_{\infty}^{-1}$ is the total weight/partition function (and
$(x;q)_{n}=\prod_{0\leq i\leq n-1}(1-xq^{i})$ is the $q$-Pochhammer symbol).
Our main result concerns the asymptotic distribution of $\lambda_{1}$ as $q\to
1-$. It is as follows.

######  Theorem 1.

Let $\lambda_{1}$, using (1), be the largest part of a width $2N$ cylindric
plane partition distributed as in (2). Then

  * •

(i) ($N$ fixed, $a$ grows critically) if
$q=e^{-\epsilon},a=e^{-\alpha\epsilon}$ as $\epsilon\to 0+$ and for $N$ fixed
we have:

$\lim_{\epsilon\to
0+}\mathbb{P}\left(\frac{\lambda_{1}-2\epsilon^{-1}\log\epsilon^{-1}}{\epsilon^{-1}}<s\right)=F^{(N)}_{\alpha,\mathrm{\,Bessel}}=\det(1-K^{(N)}_{\alpha,\mathrm{\,Bessel}})_{L^{2}(s,\infty)}$
(3)

where the determinant on the right is a Fredholm determinant of the following
finite temperature Bessel kernel (operator) in exponential coordinates

$K^{(N)}_{\alpha,\mathrm{\,Bessel}}(x,y)=e^{-\frac{x}{2}-\frac{y}{2}}\int_{0}^{\infty}\frac{1}{1+u^{N}}J_{\alpha}(2\sqrt{ue^{-x}})J_{\alpha}(2\sqrt{ue^{-y}})du$
(4)

with $J$ the Bessel function of the first kind;

  * •

(ii) ($N$ grows slowly, $a$ fixed) if $q=e^{-\epsilon},N=\beta\epsilon^{-2/3}$
as $\epsilon\to 0+$ and with $0<a<1$ fixed, we have (below
$c_{1}=-2\log(1-\sqrt{a}),c_{2}=2^{-1/3}a^{1/6}(1-\sqrt{a})^{-2/3}$):

$\lim_{\epsilon\to
0+}\mathbb{P}\left(\frac{\lambda_{1}-c_{1}\epsilon^{-1}}{c_{2}\epsilon^{-1/3}}<s\right)=F^{(\beta
c_{2})}_{\mathrm{\,Airy}}=\det(1-K^{(\beta
c_{2})}_{\mathrm{\,Airy}})_{L^{2}(s,\infty)}$ (5)

where the Fredholm determinant on the right is for the finite temperature Airy
kernel

$K^{(\beta)}_{\mathrm{\,Airy}}(x,y)=\int_{-\infty}^{\infty}\frac{e^{\beta
v}}{1+e^{\beta v}}\mathrm{\,Ai}(x+v)\mathrm{\,Ai}(y+v)dv$ (6)

with $\mathrm{\,Ai}$ the (exponentially decaying at $\infty$) Airy function.

The distributions appearing above are Fredholm determinants. Recall that an
operator $K$ on $L^{2}(X)$ and having kernel $K(x,y)$ ($X=(s,\infty)$ for us)
acts via matrix multiplication $(Kf)(x)=\int_{X}K(x,y)f(y)dy$. If $K$ is trace
class—see e.g. [21], the _Fredholm determinant_ of $1-K$ ($1$ the identity
operator) on $L^{2}(X)$ is the quantity

$\det(1-K)_{L^{2}(X)}=1+\sum_{m\geq
1}\frac{(-1)^{m}}{m!}\int_{X}\cdots\int_{X}\det_{1\leq i,j\leq
m}[K(x_{i},x_{j})]dx_{1}\cdots dx_{m}.$ (7)

### A few important remarks.

We make the following observations.

######  Remark 2.

The kernel $K^{(N)}_{\alpha,\mathrm{\,Bessel}}$ is similar to the finite-
temperature Bessel kernel recently introduced in the physics literature by
Lacroix-A-Chez-Toine et al [15], at inverse finite temperature $N$.

  * •

As $N\to\infty$ the Fermi factor $1/(1+u^{N})$ becomes the indicator
$\mathbbm{1}_{[0,1]}$ and the kernel becomes

$K^{(\infty)}_{\alpha,\mathrm{\,Bessel}}(x,y)=e^{-\frac{x}{2}-\frac{y}{2}}B_{\alpha}(e^{-x},e^{-y})$
(8)

with
$B_{\alpha}(x,y)=\int_{0}^{1}J_{\alpha}(2\sqrt{ux})J_{\alpha}(2\sqrt{uy})du$
the Bessel kernel found in the scaling of the smallest eigenvalue of random
matrix ensembles with a hard edge (e.g. the Laguerre or Jacobi ensembles
scaled around 0, see [23]). Furthermore, the probability distribution
$F^{(\infty)}_{\alpha}(s)=\det(1-K^{(\infty)}_{\alpha,\mathrm{\,Bessel}})_{L^{2}(s,\infty)}$
(already at “$0=1/(N=\infty)$ temperature”) interpolates between the standard
Gumbel distribution $(\alpha=0)$ and the Tracy–Widom distribution
($\alpha\to\infty$) [13].

  * •

As $N\to 0$ (a combinatorially meaningless limit!) we have (upon a change of
variables and using the orthogonality of Bessel $J$ functions)

$K^{(0)}_{\alpha,\mathrm{\,Bessel}}(x,y)=e^{-\frac{x}{2}-\frac{y}{2}}\delta(x-y)$
(9)

(in a distributional sense, with $\delta$ the Dirac delta measure) and
$F^{(0)}(s)=\det(1-K^{(0)}_{\alpha,\mathrm{\,Bessel}})_{L^{2}(s,\infty)}=e^{-e^{-s}}$
becomes the Gumbel distribution.

######  Remark 3.

The kernel $K^{(\beta c_{2})}_{\mathrm{\,Ai}}$ is Johansson’s [12] finite
temperature Airy kernel (at inverse temperature $\beta c_{2}$). When
$\beta\to\infty$ it becomes the usual Airy kernel (by the same argument as
above, the Fermi factor in the integrand becoming $\mathbbm{1}_{[0,\infty)}$)
with Fredholm determinant on $(s,\infty)$ the Tracy–Widom GUE distribution
[22] (the “universal” asymptotic distribution of the largest eigenvalue of
Hermitian random matrices with iid entries). A more subtle $\beta\to 0$ limit
recovers the Gumbel distribution—see [12, 3].

######  Remark 4.

As already observed in [15], one can presumably transition from our finite
temperature Bessel kernel to the finite temperature Airy kernel by using the
well-known asymptotics for Bessel functions [18, eq. 10.19.8]:

$(2/\nu)^{\tfrac{1}{3}}J_{\nu}(\nu+x\nu^{\tfrac{1}{3}})=\mathrm{\,Ai}(-2^{\tfrac{1}{3}}x)+O(\nu^{-\tfrac{2}{3}}),\quad\nu\to\infty$
(10)

but we shall not pursue this purely analytical point here. This gives rise to
the hard-to-soft edge transition in random matrix theory, see e.g. [8].

######  Remark 5.

There are other limits one can take not covered here. For example, if $q\to 1$
while $a$ and $N$ are fixed, we expect Tracy–Widom fluctuations (zero
temperature behavior). We also expect Tracy–Widom if $0<q<1$ is fixed (even if
$a\to 1$ and possibly even if $N$ is growing). More interestingly, we can take
$q,a\to 1$ as $N\to\infty$. Here the behavior is unexplored and we hope to
address this in future work. There are interesting limits as $q$ and/or $a$ go
to 0, and one of them leads to the cylindric Plancherel measure of [6] and [3,
Thm. 1.1].

######  Remark 6.

One limit that is worth mentioning as it is combinatorial is that of
$N\to\infty$ independent of $a,q$. If we take $N\to\infty$ with $a,q$ fixed
then the cylindric geometry goes away (the sides go to infinity, the seam
becomes $0$) and we look at regular trace-and-volume-weighted plane
partitions. These were covered in [5] (see in particular Remark 6 there).
Asymptotically we get either Bessel (in exponential coordinates) behavior or
Tracy–Widom behavior depending on whether $a\to 1$ or is fixed. Alternatively,
we can take $N$ to infinity after taking $a,q\to 1$ (see above), and we
recover the same behavior. The two limits commute.

######  Remark 7.

An extended version of Thm. 1 is also possible. That it, if one looks at the
top $k$ lozenges in a neighborhood of the peak, the joint distribution of
those is also asymptotically a Fredholm determinant, this time of a “time-
extended” kernel. In the Bessel limit perhaps this is not too interesting as
“time” (partition number around 0 in (1)) remains discrete. In the continuous
case though a rescaling is needed and one would then obtain, as was done in
[3], the finite temperature/periodic Airy process of [16].

######  Remark 8.

Let us notice one last thing: for certain parameter ranges, our main theorem
combined with recent results of [1] could perhaps elucidate certain (perhaps
expected) behavior of maxima of Gaussian free fields on a cylinder.

Figure 1: Left: a cylindric plane partition of width $2N=6$, seam $3$, trace
$11$, volume $44$, and a peak of $6$. Right: a depiction of last passage
percolation on a tie. Each horizontal row $i$ of lozenges has iid
$Geom(aq^{i})$ random variables inside each lozenge. The longest down-
right/left path has length $L=53$.

##  2 Last passage percolation and $q$-Whittaker measures

In this section we connect the above results with a model of directed last
passage percolation on a cylinder. Similar models were studied in [4] for the
case of reflecting strips (equivalently, vertically symmetric cylinders), and
while the combinatorial tools apply mutatis-mutandis, the probability measures
studied are different.

Fix numbers $0<a,q<1$ (for simplicity). Let us consider the following so-
called last passage percolation on a tie (see also [4]). In Figure 1 (right)
the two vertical sides are identified, so we are on a cylinder. The cylinder,
of width $2N$, is tiled with $N\times N$ big squares (with bolded edges, not
so important for what follows), and each such big square contains $N^{2}$ unit
squares. In each unit square we have a geometric random variable, and they are
all independent of one another. On horizontal slice $i\geq 1$, where a slice
is a sequence of squares touching only at the corners and sharing a common
horizontal line through their centers, the random variables are iid ${\rm
Geom}(aq^{i})$. (A random variable $X$ is ${\rm Geom}(t)$ if
$\mathbb{P}(X=k)=(1-t)t^{k}$. Note that slice $i$ contains $\min(i,N)$
squares.)

In the geometry described above, consider paths going from the top-most unit
square down to $\infty$ and using only unit down-right or down-left steps. The
length of a path is the sum of the random geometric integers it encounters.
Now consider the longest such path, and call the length $L$. By
Borel–Cantelli, the cylinder contains only a finite number of squares whose
random variables are non-zero, and thus $L<\infty$ almost surely. We then have
the following theorem.

######  Theorem 9.

It holds that

$L+\chi=\lambda_{1}\quad\text{in\ distribution}$ (11)

where $\lambda_{1}$ is the peak of a cylindric plane partition distributed
according to (2), $L$ is as above, and $\chi$ is a random variable independent
of $L$ with explicit distribution $\mathbb{P}(\chi\leq
m)=(q^{N(m+1)};q^{N})_{\infty}$. (In fact, $\chi=\kappa_{1}$ is the
distribution of the first part of a random integer partition $\kappa$ with
$\mathbb{P}(\kappa)=(q^{N};q^{N})^{-1}_{\infty}(q^{N})^{|\kappa|}$).

######  Proof 2.1.

The proof is the same as that of Thm. 2.1 of [4], with the exception that
there the sides of the tie were reflecting, and here they are identified. In
other words free boundary conditions are replaced by periodic boundary
conditions. Note the proof uses nothing more than the classical
Robinson–Schensted–Knuth correspondence [14] and Greene’s theorem [10],
adapted to periodic/free boundary conditions.

Let us state another important remark. In recent work of
Imamura–Mucciconi–Sasamoto [11] it was proven that $L$ (via $\lambda_{1}$) is
related to an observable of a so-called $q$-Whittaker measure. The
$q$-Whittaker symmetric polynomials $P_{\rho}(x;q)$ and $Q_{\rho}(x;q)$ are
the Macdonald $(q,t)$ symmetric polynomials [17] in variables
$x=(x_{1},\dots,x_{N})$ at $t=0$. What was proven in [11] together with the
above yields the following result.

######  Proposition 10.

One has $L+\chi=\rho_{1}+\chi^{\prime}$ in distribution where $\rho_{1}$ is
the first part of a random partition distributed with the $q$-Whittaker
measure

$\mathbb{P}(\rho)=\tilde{Z}^{-1}P_{\rho}(\sqrt{a}q^{\frac{1}{2}},\dots,\sqrt{a}q^{N-\frac{1}{2}};q^{N})Q_{\rho}(\sqrt{a}q^{\frac{1}{2}},\dots,\sqrt{a}q^{N-\frac{1}{2}};q^{N})$
(12)

with $\tilde{Z}=\prod_{1\leq i,j\leq N}(q^{i+j-1};q^{N})_{\infty}^{-1}$ (each
polynomial has $N$ variables $x_{i}=\sqrt{a}q^{i-\tfrac{1}{2}},1\leq i\leq
N$), and $\chi,\chi^{\prime}$ are independent of everything with the same
distribution as in Thm. 9 $\mathbb{P}(\chi\leq
m)=(q^{N(m+1)};q^{N})_{\infty}$.

The importance of such measures was first realized by Borodin–Corwin [7] who
showed that $\rho_{1}$ (for $\rho$ $q$-Whittaker distributed for certain
specializations of symmetric functions _different from the ones above_)
converges to the height function of the so-called stochastic heat equation (a
relative of the Kardar–Parisi–Zhang or KPZ equation).

Our main asymptotic result of this paper together with the discussion in this
section seem to then suggest that one can also see the hard edge of random
matrix ensembles in the long-time behavior of KPZ models. Notice that the
finite temperature Airy kernel is already known to appear in the solution of
the KPZ equation at finite time with wedge initial conditions. See for example
[2]. We plan to investigate this further in future work.

##  3 Proof of Theorem 1

We now attempt to sketch a proof of Theorem 1. We start from the measure (2).
Via the representation of (1), it is not hard to see that the measure can be
rewritten as

$\mathbb{P}(\Lambda)=Z^{-1}(q^{N})^{|\mu|}\prod_{-N\leq
i\leq-1}s_{\lambda^{(i+1)}/\lambda^{(i)}}(\sqrt{a}q^{|i|-1/2})\prod_{1\leq
i\leq N}s_{\lambda^{(i-1)}/\lambda^{(i)}}(\sqrt{a}q^{i-1/2})$ (13)

where we recall $\lambda^{(\pm N)}=\mu,\lambda^{(0)}=\lambda$ by definition.
Here $s_{\lambda/\mu}$ are the skew Schur polynomials [17, Ch. 1], and they
are evaluated in one variable, so the above is immediate as
$s_{\lambda/\mu}(x_{1})=x_{1}^{|\lambda|-|\mu|}\mathbbm{1}_{\mu\prec\lambda}$.
By the branching rule we have that the $\lambda$ marginal is

$\mathbb{P}(\lambda)\propto\sum_{\mu}q^{N|\mu|}\left[s_{\lambda/\mu}(\sqrt{a}q^{\tfrac{1}{2}},\dots,\sqrt{a}q^{N-\tfrac{1}{2}})\right]^{2}$
(14)

and it turns out such measures, called cylindric or periodic Schur measures,
were studied before by Borodin (who also derived bulk asymptotics for periodic
plane partitions) and Betea–Bouttier (who derived edge asymptotics for a
simpler periodic Plancherel measure).

The point process associated to the partition $\lambda$, that is the set
$\\{\lambda_{i}-i+1/2\\}$, is nonetheless not determinantal (despite a certain
determinantal structure already present above). However, the _shift-mixed or
grand canonical_ point process $\\{\lambda_{i}-i+1/2+c\\}$, where $c$ is a
random variable independent of everything else with explicit distribution
$\mathbb{P}(c)=t^{c}q^{Nc^{2}/2}/\theta_{3}(t;q^{N})$ is indeed determinantal,
with $t$ a parameter which we can fix to be 1 throughout111This shift is
sometimes called “passage to the grand canonical ensemble” in the physics
literature.. (Here
$\theta_{3}(t;u):=\sum_{c\in\mathbb{Z}}t^{c}u^{c^{2}/2}=(u;u)_{\infty}(-\sqrt{u}z;u)_{\infty}(-\sqrt{u}/z;u)_{\infty}$
is one of the Jacobi theta functions.) Precisely it means that if
$\\{k_{1},\dots,k_{n}\\}\in\mathbb{Z}+\tfrac{1}{2}$ is a finite set of
positions, the following determinantal identity holds [6, 3]:

$\mathbb{P}(\\{k_{1},\dots,k_{n}\\}\in\\{\lambda_{i}-i+1/2+c\\})=\det_{1\leq
i,j\leq n}K(k_{i},k_{j})$ (15)

where $K$ is the following discrete kernel/operator (below we follow notation
from [3]):

$K(k,k^{\prime})=\frac{1}{(2\pi
i)^{2}}\int_{|z|=1+\delta}\frac{dz}{z^{k+1}}\int_{|w|=1-\delta}\frac{dw}{w^{-k^{\prime}+1}}\cdot\frac{F(z)}{F(w)}\cdot\kappa(z,w)$
(16)

where $\delta$ is a small positive number and

$F(z):=\frac{(\sqrt{a}q^{1/2}/z;q)_{\infty}}{(\sqrt{a}q^{1/2}z;q)_{\infty}},\quad\kappa(z,w):=\sqrt{\frac{w}{z}}\cdot\frac{(q^{N};q^{N})^{2}_{\infty}}{\theta_{q^{N}}(w/z)}\cdot\frac{\theta_{3}(t\frac{z}{w};q^{N})}{\theta_{3}(t;q^{N})}.$
(17)

(Here $\theta_{u}(x):=(x;u)_{\infty}(u/x;u)_{\infty}$ is another theta
function, in multiplicative notation.) The contour integrals above signify
coefficient extraction, and make sense analytically (this becomes important
for asymptotic analysis!) as long as the $z$ contour is outside the $w$
contour and both are very close to the unit circle222A sufficient set of
conditions has i) $1<\frac{|z|}{|w|}=\frac{1+\delta}{1-\delta}<q^{-N}$; ii)
$1+\delta=|z|<(aq)^{-1/2}$ and iii) $1-\delta=|w|>(aq)^{1/2}$—see [3]..

The principle of inclusion-exclusion then gives that the distribution of
$L=\lambda_{1}-1/2+c$ is a discrete Fredholm determinant of $K$:
$\mathbb{P}(L\leq\ell)=\det(1-K)_{l^{2}\\{\ell+1/2,\ell+3/2,\dots\\}}$.

We now show, in the scaling regime of part (i), that
$\det(1-K)\to\det(1-K^{(N)}_{\alpha,\mathrm{\,Bessel}})$ in the limit
$\ell=\epsilon^{-1}s+2\epsilon^{-1}\log\epsilon^{-1}$ as $\epsilon\to 0+$. We
only show that $\epsilon^{-1}K(k,k^{\prime})\to
K^{(N)}_{\alpha,\mathrm{\,Bessel}}(x,y)$ for
$(k,k^{\prime})=\epsilon^{-1}(x,y)+2\epsilon^{-1}\log\epsilon^{-1}$. One then
further shows, using a similar argument, that $\mathrm{tr}\,K$ on
${l^{2}\\{\ell+\tfrac{1}{2},\ell+\tfrac{3}{2},\dots\\}}$ converges to
$\mathrm{tr}\,K^{(N)}_{\alpha,\mathrm{\,Bessel}}$ on $L^{2}(s,\infty)$ to
conclude. There are analytic justifications needed for interchanging limits
and integrals, based on dominated convergence, that are implicit everywhere
and which we omit.

Let us then prove $\epsilon^{-1}K(k,k^{\prime})\to
K^{(N)}_{\alpha,\mathrm{\,Bessel}}(x,y)$ in the regime above-described. First
we use the fact that the infinite-length $q$-Pochhammer symbol (the $q$-Gamma
function) converges to the Gamma function:

$\log(q^{c};q)_{\infty}=-\frac{\pi^{2}}{6}\epsilon^{-1}+\left(\frac{1}{2}-c\right)\log\epsilon+\frac{1}{2}\log(2\pi)-\log\Gamma(c)+O(\epsilon)$
(18)

with $q=e^{-\epsilon},\epsilon\to 0+,c\notin-\mathbb{N}$. To keep the discrete
kernel $\epsilon^{-1}K(k,k^{\prime})$ finite we change variables
$(z,w)=(e^{\epsilon\zeta},e^{\epsilon\omega})$ and notice the contours
transform to vertical up-oriented lines close to the imaginary axis, the
$\zeta$ on the right and the $\omega$ on the left. Finally in this limit we
can use Lemma 5.5 of [3] to estimate $\kappa(z,w)\sim\frac{\pi}{\epsilon
N\sin\frac{(\zeta-\omega)}{N}},\epsilon\to 0+$. Putting this all together,
$\epsilon K(k,k^{\prime})$ converges to the following kernel:

$(x,y)\mapsto\frac{1}{(2\pi
i)^{2}}\int_{i\mathbb{R}+\eta}d\zeta\int_{i\mathbb{R}-\eta}d\omega\frac{f(\zeta)e^{-x\zeta}}{f(\omega)e^{-y\omega}}\frac{\pi}{N\sin\frac{(\zeta-\omega)}{N}},\quad
f(\zeta):=\frac{\Gamma(\frac{\alpha+1}{2}+\zeta)}{\Gamma(\frac{\alpha+1}{2}-\zeta)}$
(19)

(for some $\eta>0$ small enough). To arrive at the stated form, we first
notice
$\frac{\pi}{N\sin\frac{(\zeta-\omega)}{N}}=\int_{-\infty}^{\infty}\frac{e^{(N+\omega-\zeta)v}dv}{1+e^{Nv}}$.
In the triple integral we observe Bessel functions appearing explicitly via
[18, eq. 10.19.22] $J_{\alpha}(X)=\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}\frac{\Gamma(-Z)(X/2)^{\alpha+2Z}dZ}{\Gamma(\alpha+1+Z)}$
(with $Z=\zeta-\frac{\alpha+1}{2},X=2\sqrt{e^{-x-v}}$, and similarly for
$\omega$ and $y$). A final substitution $u=e^{-v}$ gives the stated form.

Finally, so far we obtained asymptotics for the shifted quantity
$\lambda_{1}+c-1/2$, but the difference, after scaling, is $\epsilon(c-1/2)$
which converges to 0 in probability so the shift can be removed and part (i)
follows.

We now turn to the proof of part (ii), and the only difference with the above
is the asymptotic analysis. Let us first put $R=\epsilon^{-1}\to\infty$ and
$b=\sqrt{a}$ for notational convenience. The strategy is the same, but now we
will prove we have $R^{1/3}K(k,k^{\prime})\to K^{(\beta
c_{2})}_{\mathrm{\,Airy}}(x,y)$ with $K$ the discrete kernel from above and
with $(k,k^{\prime})=c_{1}R+(x,y)c_{2}R^{1/3}$ where we recall
$c_{1}=-2\log(1-b),c_{2}=2^{-1/3}b^{1/3}(1-b)^{-2/3}$.

As $0<b<1$ is fixed, we can use the estimate

$(Bq;q)_{\infty}\sim-\epsilon^{-1}\mathrm{Li}_{2}(B),\quad
q=e^{-\epsilon},\quad\epsilon\to 0+$ (20)

(if $B$ is away from $0$ and $1$). Here $\mathrm{Li}_{2}$ is the dilogarithm
function. Asymptotically then we estimate $R^{1/3}K(k,k^{\prime})$ to be

$\frac{R^{1/3}}{(2\pi
i)^{2}}\oint\oint\frac{e^{R(S(z)-S(w))}}{z^{k_{0}+1}w^{-k^{\prime}_{0}+1}}\cdot\kappa(z,w)dzdw$
(21)

where $S(z)=\mathrm{Li}_{2}(bz)-\mathrm{Li}_{2}(b/z)-c_{1}\log z$ and
$(k_{0},k^{\prime}_{0})=(k,k^{\prime})-c_{1}R$. The main contribution of this
integral then comes from the double critical point of the action $S$, and the
argument as in [3, Prop. 5.1] goes through with little modification. The
double critical point of $S$ is at $z=1$, and to see a finite contribution of
the action we rescale $z$ and $w$ zooming in around 1 as $(z,w)=(e^{\zeta
R^{-1/3}/c_{2}},e^{\omega R^{-1/3}/c_{2}})$. The action $S$ has a Taylor
expansion around 1 given by

$S(1)+\frac{S^{\prime\prime\prime}(1)}{6}\frac{\zeta^{3}}{c_{2}^{3}}-x\zeta+O(R^{-1/3})=S(1)+\frac{\zeta^{3}}{3}-x\zeta+O(R^{-1/3})$
(22)

where $c_{2}$ was chosen explicitly to cancel $S^{\prime\prime\prime}(1)$.
Furthermore, let us note that as $N=\beta R^{2/3}$ we have $q^{N}=e^{-\beta
R^{-1/3}}$ (this is the “important” parameter in $\kappa(z,w)$) and so again
we can estimate $\kappa(z,w)\sim\frac{\pi}{R^{1/3}\beta
c_{2}\sin\frac{\zeta-\omega}{\beta c_{2}}}$.

We then arrive, as $R\to\infty$, to the following limiting kernel for
$R^{1/3}K(k,k^{\prime})$:

$(x,y)\mapsto\frac{1}{(2\pi
i)^{2}}\int_{i\mathbb{R}+\eta}d\zeta\int_{i\mathbb{R}-\eta}d\omega\frac{e^{\zeta^{3}/3-x\zeta}}{e^{\omega^{3}/3-y\omega}}\frac{\pi}{\beta
c_{2}\sin\frac{\zeta-\omega}{\beta c_{2}}}$ (23)

(for some $\eta$ small enough). We use the same integral trick as in part (i)
above to transform $\frac{\pi}{\beta c_{2}\sin\frac{\zeta-\omega}{\beta
c_{2}}}$ and we recognize that $\mathrm{\,Ai}(x)=\frac{1}{2\pi
i}\int_{\eta+i\mathbb{R}}e^{\zeta^{3}/3-x\zeta}d\zeta$. This finally leads to
the stated form of $K^{(\beta c_{2})}_{\mathrm{\,Airy}}(x,y)$ and modulo the
same analytical arguments omitted above, we conclude the proof.

##  4 Conclusion

In this work, using a mixture of algebraic combinatorics, mathematical
physics, and asymptotic analysis, we analyzed certain asymptotic distributions
of peaks of simple cylindric plane partitions. Our work is far from exhaustive
and builds on work of the authors [5] and of Betea–Bouttier [3], and these
works already build and expand upon work of Okounkov [19],
Okounkov–Reshetikhin [20], and Borodin [6].

We identify four further directions worth investigating, besides what was
already commented on in the remarks in the introduction. One is to consider
cylindric plane partitions with more complex features (e.g. different
profiles, more than one peak, etc.), or indeed other types of tilings like
domino tilings, and to study their “edge” asymptotic behavior. We plan to
address this in future work. The second is to make our treatment of last
passage percolation on a cylinder/reflecting strip systematic, and this is
current joint work in progress of the first author with J. Bouttier, P.
Nejjar, and M. Vuletić. Thirdly, we would like to explore possible connections
to the stochastic heat equation and the KPZ equation in more detail, via the
bridge built in [11] and the partition function of $\log-\Gamma$ polymers.
Fourth, and as results of [5] show, it should be easy to obtain finite
temperature analogues of other random matrix-type kernels like the Meijer-G
kernel for example. We note however that we lose a direct connection to
Muttalib–Borodin ensembles that was present in [5].

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11institutetext:  1Space Research Institute, Russian Academy of Sciences,
Profsoyuznaya 84/32, 117997 Moscow, Russia
2Department of Physics and Astronomy, FI-20014 University of Turku, Finland
3Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1,
D-72076 Tübingen, Germany

#  SRG/ART-XC discovery of SRGA J144459.2$-$604207:
a well-tempered bursting accreting millisecond X-ray pulsar

Sergey V. Molkov 11 Alexander A. Lutovinov 11 Sergey S. Tsygankov 22 Valery F.
Suleimanov 33 Juri Poutanen 22 Igor Yu. Lapshov 11 Ilia A. Mereminskiy 11
Andrei N. Semena 11 Vadim A. Arefiev 11 Alexey Yu. Tkachenko 11

We report on the discovery of the new accreting millisecond X-ray pulsar SRGA
J144459.2$-$604207 using the SRG/ART-XC data. The source was observed twice in
February 2024 during the declining phase of the outburst. Timing analysis
revealed a coherent signal near 447.8 Hz modulated by the Doppler effect due
to the orbital motion. The derived parameters for the binary system are
consistent with the circular orbit with a period of $\sim 5.2$ h. The pulse
profiles of the persistent emission, showing a sine-like part during half a
period with a plateau in between, can well be modelled by emission from two
circular spots partially eclipsed by the accretion disk. Additionally, during
our 133 ks exposure observations, we detected 19 thermonuclear X-ray bursts.
All bursts have similar shapes and energetics, and do not show any signs of
photospheric radius expansion. The burst rate decreases linearly from one per
$\sim$1.6 h at the beginning of observations to one per $\sim$2.2 h at the end
and anticorrelates with the persistent flux. Spectral evolution during the
bursts is consistent with the models of the neutron star atmospheres heated by
accretion and imply a neutron star radius of 11–12 km and the distance to the
source of 8–9 kpc. We also detected pulsations during the bursts and showed
that the pulse profiles differ substantially from those observed in the
persistent emission. However, we could not find a simple physical model
explaining the pulse profiles detected during the bursts.

###### Key Words.:

pulsars: individuals: SRGA J144459.2$-$604207 – stars: neutron – X-rays:
binaries – X-rays: bursts

##  1 Introduction

Accreting millisecond X-ray pulsars (AMXPs) constitute a relatively small
group of binary systems featuring a rapidly rotating neutron star (NS) and a
low-mass optical companion (see Patruno & Watts 2021; Di Salvo & Sanna 2022,
for recent reviews). The NSs in these systems, believed to be progenitors of
rotation-powered millisecond pulsars, exhibit spin periods ranging from $\sim
1.7$ to $\sim 10$ ms and possess relatively weak magnetic fields (around
$10^{8}$–$10^{9}$ G). Thus these object play an important role in the star
evolutionary processes, but the current sample of known AMXPs comprises only
about two dozen sources. Therefore search for such objects and their
discoveries are quite important task. Moreover this task is very non-trivial
one and demands extraordinary technical capabilities of X-ray instruments,
including a large effective area and high time resolution.

The Mikhail Pavlinsky ART-XC telescope (Pavlinsky et al. 2021) onboard the
Spectrum-Roentgen-Gamma observatory (SRG; Sunyaev et al. 2021) discovered a
new AMXP, SRGA J144459.2$-$604207 (hereafter SRGA J1444), during the ongoing
all-sky survey. The source was found on 2024 Feb 21 at the position close to
the Galactic plane with coordinates ($l,b$) =
($316\aas@@fstack{\circ}4,-0\aas@@fstack{\circ}8$) and the flux of $\sim 100$
mCrab in the 4–12 keV energy band (Mereminskiy et al. 2024).

The intense follow-up campaign carried out immediately after the discovery
revealed that SRGA J1444 is a new accreting millisecond pulsar with a spin
period of 447.8 Hz (Ng et al. 2024) showing regular Type-I X-ray bursts
(Mariani et al. 2024; Ray et al. 2024; Sanchez-Fernandez et al. 2024).
Subsequent observations of SRGA J1444 with NICER and Insight-HXMT instruments
unveiled a clear sinusoidal Doppler shift of the spin frequency that allowed
to determine an orbital period of $\sim$5.2 h, indicating a companion star
mass exceeding 0.255 M⊙ (Ray et al. 2024; Li et al. 2024).

The improved coordinates of SRGA J1444, RA (J2000) =
14h44m58$\aas@@fstack{s}$9, Dec (J2000) =
$-$60°41′55$\aas@@fstack{\prime\prime}$3, were obtained with the High
Resolution Camera on board the Chandra observatory (Illiano et al. 2024).
Despite the accurate localization of the source, no optical/IR counterparts
were found (Sokolovsky et al. 2024; Cowie et al. 2024; Baglio et al. 2024;
Saikia et al. 2024). A radio counterpart was discovered at the position of
SRGA J1444 using the Australia Telescope Compact Array (Russell et al. 2024).
Its spectral index consistent with the emission from either a compact jet or
discrete ejecta from an X-ray binary.

Interestingly, that the retrospective analysis of the MAXI and INTEGRAL
archival data revealed past X-ray activity of SRGA J1444. Particularly, an
increase in the flux from the sky position coincident with the SRGA J1444 one
was observed in the beginning of Jan 2022 and mid-Dec 2023 (Negoro et al.
2024; Sguera & Sidoli 2024).

In this paper, we report on the discovery of a new AMXP SRGA J1444 using the
ART-XC telescope data. Results of timing and spectral analysis both the
persistent emission and its evolution during multiple Type-I bursts are
presented as well as modelling of the bursts and pulse profile.

##  2 Observations and data reduction

Figure 1: Temporal evolution of the SRGA J1444 emission during the 2024
outburst. Upper panel: the ART-XC light curve during two observations. For
type-I X-ray bursts their time and mean flux are shown with the dashed line
and thick streak. Bottom left: The flux evolution as observed by MAXI/GSC in
the 4–10 keV energy band. Bottom right: the X-ray burst recurrence time (in
black, left axis) and the mean persistent flux between the bursts in the 4–25
keV energy band $F_{\rm 4-25}^{\rm pst}$ (in red, right axis) as measured by
ART-XC.

SRGA J1444 was first detected in the data downlinked from the ART-XC telescope
on 2024 Feb 21 using a near-real-time data processing chain. It was quickly
identified as a new X-ray source located in the Galactic plane. At the time of
the detection the source flux was about $(1.9\pm 0.1)\times 10^{-9}$ erg cm-2
s-1in the 4–12 keV band (Mereminskiy et al. 2024, see also Fig. 1). Two
extended observations were conducted with the ART-XC telescope as part of the
follow-up program. The first observation began on 2024 Feb 2 and lasted for 28
ks, while the second observation started on 2024 Feb 26 and continued for 105
ks.

The ART-XC telescope is an imaging instrument consisting of seven modules and
operating in the photon counting mode (Pavlinsky et al. 2021) in the 4–30 keV
energy band. The data were processed using the artproducts v1.0 software with
the latest CALDB v20220908. For both spectral and timing analysis we extracted
photons from the circle of 1$\aas@@fstack{\prime}$8 radius around the source
position applying also an energy filtering and excluding high energies ($>25$
keV). Due to the limited energy band of ART-XC, which does not permit a
reliable determination of a moderate interstellar absorption, for energy
spectra fitting we employed models with the fixed equivalent hydrogen column
density of $2.7\times 10^{22}$ cm-2, determined from the NICER data (Ng et al.
2024). All reported flux values in the paper are unabsorbed unless explicitly
stated otherwise.

##  3 Persistent emission and bursting behavior

Figure 2: Average energy spectrum of all type-I X-ray bursts observed by ART-
XC during first observation (blue crosses) and the spectrum of the persistent
emission extracted between bursts number II and III (red crosses). The pink
curves represent the spectral model of the absorbed blackbody and the cutoff
power law, respectively.

Observations of SRGA J1444 with the ART-XC telescope were carried out soon
after the maximum of the outburst at the declining phase (Fig. 1, lower left
panel). In total, in two observations, a total of 19 Type-I X-ray bursts were
registered, each lasting about 45 s and occurring at approximately equal
intervals of about two hours. However, a detailed study showed that the
recurrent time $\Delta t$ between bursts is not constant, and the burst rate
decreases linearly from one per $\sim$1.6 h at the beginning of observations
to one per $\sim$2.2 h at the end (Fig. 1, lower right panel), while the
energy release during the bursts remains approximately at the same level (Fig.
1, upper panel).

To trace the evolution of the persistent emission, we split our observations
into intervals of about 800 s, with X-ray bursts being excluded from the
analysis. Then we determined the average count rate in the 4–25 keV energy
band for each interval. The resulting light curve in Crab flux units is
presented on the upper panel of Fig. 1.

To quantify the rate of change in the persistent flux and establish its
relationship with the burst generation frequency, we first reconstructed the
energy spectra during intervals between the bursts. Then we fitted these
spectra with a simple analytical model consisting of the absorbed power law
with a high energy cutoff, tbabs$\times$cutoffpl in xspec (Arnaud 1996), and
estimated the corresponding fluxes. All spectra have a similar shape and are
well approximated by this model with the photon index of $\Gamma\sim 1.8$ and
high energy cutoff energy $E_{\rm cut}\sim 25$ keV and differ only by their
normalizations. As an example, one of the spectra of the persistent emission
extracted between the bursts II and III is shown in Fig. 2. The evolution of
the flux between the bursts is presented on the lower right panel in Fig. 1
(in red) and it is clearly seen that this flux is anticorrelated with the
bursts recurrence time. At the same time, the total energy released between
bursts in the 4–25 keV energy band remains approximately constant, with the
fluence being $E_{\rm bet\\_brst}\simeq 1.48\times 10^{-5}$ erg cm-2.

###  3.1 Type-I X-ray bursts

Figure 3: Light curves of all detected type-I X-ray bursts in the 4–25 keV
energy band. The bottom panel shows the averaged burst profile. Dashed
vertical lines represent the time intervals used for the time-resolved
spectral analysis (see Sect. 3.2).

First of all, the averaged energy spectrum for each of the bursts was
analysed. We reconstructed ART-XC spectra in the 4–25 keV energy band
collected during 45 s time intervals starting from the burst beginning. Note,
that the total spectrum of the source during the burst consists of the
persistent emission and the spectrum of a thermonuclear flash itself. The
contribution of the persistent emission to the total burst spectrum was
estimated by constructing the energy spectrum of the source during 800 s
preceding the burst. All pre-burst spectra are well approximated by the cut-
off power-law model and the same parameters within the uncertainties, as for
the persistent emission (see Sect. 3). In order to describe the energy spectra
of thermonuclear bursts, we approximated them with the two-components model,
including a blackbody and a cut-off power law (modified by interstellar
absorption with fixed equivalent hydrogen column density, see above). The
parameters of the second component, excluding normalization, were fixed at the
values obtained for the persistent emission.

This model describes all 19 bursts relatively well with the same within errors
blackbody flux of $F_{4-25}\simeq 8\times 10^{-9}$ erg s-1 cm-2 and the
temperature, $kT_{\rm bb}\simeq 2.3$ keV (see Fig. 2). The fluence released
per burst in the 4–25 keV energy band is about $E_{4-25}^{\rm bst}\simeq
3.6\times 10^{-7}$ erg cm-2.

We investigated also morphology of the bursts and found that all the bursts
have a similar shape with the fast rise that transforms into a plateau-like
phase which in turn changes to an exponential decay (see upper part of Fig.
3). The similarity of the burst profiles and their spectral parameters gives
us the opportunity to analyze not each burst individually, but their sum,
which significantly improves statistical errors and allows us to study
emission properties with the better time resolution. In the lower panel of
Fig. 3, we present an average burst profile with a time resolution of 1 s.
Below, we will take a closer look at the spectral evolution during the bursts.

###  3.2 Time resolved spectral analysis of the bursts

Figure 4: Time resolved spectroscopy of the averaged X-ray bursts emission.
The blue dotted line on the bottom panel shows the count rate light curve in
the 4–25 keV with subtracted pre-burst rate and renormalized in amplitude.

In order to investigate the spectral evolution of the radiation during X-ray
bursts, we divided the averaged burst profile into 9 time intervals (see the
bottom panel of Fig. 3) and reconstructed energy spectra for them. The
contribution of the persistent emission was taken into account in the same way
as described above, i.e. using it as background for approximating the bursts
spectra. To describe these spectra, we used the blackbody model (modified by
the fixed interstellar absorption), which adequately describes the spectra for
all time intervals. The evolution of the blackbody parameters during the
bursts is shown in Fig. 4. The blackbody normalization $K_{\rm bb}$ was
converted to the radius of the emission region for the distance of 10 kpc:
$R_{\rm D10}=\sqrt{K_{\rm bb}}$. Also, in the lower part of the figure, we
showed the flux evolution in the energy range of 4–25 keV. Note that there are
no signatures of a photospheric radius expansion during the bursts. The burst
peak bolometric luminosity, $L_{\rm pk}=3.2\times 10^{38}$ erg s-1 is reached
at about tenth second, and the total energy release during the burst is
approximately $E_{\rm bst}=5.1\times 10^{39}$ erg (again the 10 kpc distance
is assumed).

##  4 Timing properties

In this section we investigate timing properties of SRGA J1444, including
pulsations, orbital motion as well as an evolution of pulse profile with the
time and energy both for the persistent emission and during the type-I X-ray
bursts.

Figure 5: Variations of the measured frequency of the NS rotation due to an
orbital motion in the binary system according to the ART-XC data (black
points). The model is shown with the red dashed-line.

###  4.1 Orbital parameters and coherent timing

A coherent signal near 447.8 Hz from SRGA J1444 was discovered by NICER (Ng et
al. 2024) and later parameters of the binary motion had been preliminary
determined based on the long set of the NICER observations (Ray et al. 2024).
Pulsations along with the sinusoidal modulation of its frequency with time are
also clearly detected in the ART-XC data. To determine the spin period and
orbital parameters, our observations were divided into segments of
approximately 800 s for the subsequent analysis (type-I X-ray bursts were
excluded). Power spectra were then computed for each data segment to measure
the frequency associated with this spin period. The resulting spin-frequency
evolution is presented in Fig 5. Further, assuming a circular motion, we
obtain the following orbital parameters for the binary system: $P_{\rm
orb}=0.217649(5)$ d is the orbital period, the projected semi-major axis is
$a_{\rm x}\sin i=0.6513(2)$ lt-s and $T_{\rm asc}=$MJD 60361.64126(5) is the
time of passage through the ascending node. These values are in good agreement
with the parameters obtained from the NICER (Ray et al. 2024) and Insight-HXMT
(Li et al. 2024) data. Subsequently, we corrected the arrival times of photons
for the orbital motion using the obtained orbital parameters. Following this
correction, we re-evaluated the NS rotation period for each of the time
segments.

We found that the measured spin period evolves with time in a complex way,
which may indicate both that our orbital solution is not accurate enough and
that the period does change, for example, due to variations in the mass
accretion rate. At first, we folded the data for each time segment with the
corresponding spin period with the same epoch time ($T_{\rm epoch}=$MJD
60361.0). The resulting segment-by-segment folding is presented on the upper
panel of Fig. 6. We see that the pulsations are reliably detected but their
phase floats from segment to segment. For further phase-resolved analysis, it
is necessary either to build a more accurate model of orbital motion and take
into account possible intrinsic spin variations in time, or to use the
existing model for individual segments, but simply introduce a correction for
phase shift for each segment. The former, i.e. the proper phase-connected
analysis, is much more complicated and, among other things, implies a well-
calibrated or well-modeled on-board clock for the entire observation time,
which in our case has not yet been done. The second approach is much simpler,
but at the same time allows to do a full-fledged phase-resolved analysis.
Therefore, we followed the second way and attributed to each segment not only
the corresponding period, but also the phase shift necessary for phase
alignment. To determine the shifts we folded the data for each interval with
the corresponding period starting from the epoch time ($T_{\rm epoch}=$MJD
60361.0) and approximated the resulted profile with the simple sine model.
Using the phase values at which the sine turns to zero, we aligned the
solutions for the intervals so that the sine equals a zero value at phase
zero. Folding the data using this “phase-aligned model” is shown in the bottom
panel of Fig. 6. Below, this model will be used for the reconstruction of
pulse profiles for the persistent emission.

Figure 6: Upper panel: Light curves for each of segments folded with period
determined for this interval assuming the same epoch time ($T_{\rm epoch}=$MJD
60361.0). Bottom panel: Phase aligned pulse profiles for all segments.

Figure 7: Evolution of the pulse profile (panels a and b) and pulsed fraction
(panel c) with the source intensity.

###  4.2 Pulse profile evolution

####  4.2.1 Pulse profiles of the persistent emission

As the first step, we investigate how the pulse profile is changing with the
flux for the persistent emission. To do this, we used 17 time intervals
between successive type-I X-ray bursts and folded the 4–25 keV source light
curve using the phase-aligned model (see Sect. 4.1). The folded curves are
shown in Fig. 7a. Using a standard expression of
$PF=(C_{\max}-C_{\min})/(C_{\max}+C_{\min})$, where $C$ is a bin count rate,
we estimated the pulsed fraction for each of the intervals and traced its
value with the flux. The dependence of the pulsed fraction on the flux in the
energy range of 4–25 keV is shown in Fig. 7c. We see that the PF is consistent
with being constant at low and medium fluxes and decreases at high fluxes. It
follows from Fig. 7a that the pulse profile in a wide energy range have only
minor difference. Particularly, at lower fluxes, it can be described by a sine
wave at phases 0.0–0.5 and a plateau in the range of 0.5–1.0. At higher
fluxes, a small bump (interpulse) appears instead of the plateau. To
demonstrate the difference more clearly, we have made pulse profiles for three
flux levels (Fig. 7b). The smaller pulsating fraction in “high state” is
probably due to the presence of the interpulse.

At the next step, an evolution of the pulsed emission with the energy was
investigated. We divided the 4–25 keV energy range of the ART-XC telescope
into 4 bands (4–8, 8–12, 12–17, and 17–25 keV) and reconstructed light curves
for all observations in each band, excluding data intervals containing X-ray
bursts. We also considered an additional channel of 25–35 keV (technically the
telescope mirrors can focus photons with the energy of up to 35 keV). In a
normal situation, this range is not used due to the low efficiency of the
events registration and the difficulty with absolute flux calibrations,
however, for qualitative analysis, it can be quite representative. Resulting
pulse profiles in different energy bands are presented in Fig. 8. It is
clearly seen that the pulse shape depends weakly on the energy. Additionally,
in all energy ranges, the “sine-like” wave part is present, while the shape of
the “plateau” is varied. There is also a small but noticeable phase shift
$\Delta\phi$ with energy, with the harder photons arriving earlier than the
softer ones, e.g., $\Delta\phi=-0.05$ for the 17–25 keV photons relative to
4–8 keV photons (i.e. the slope of the linear relation of $\Delta\phi(E)$ is
$-0.033$ keV-1).

Figure 8: Pulse profiles of the persistent emission from the source in five
energy bands. Blue lines are the approximations with a sine wave in the phase
interval 0.0–0.5.

####  4.2.2 Pulse profiles during X-ray bursts

To track the possible evolution of the pulse profile during the bursts, we
divided the bursts into 4 time intervals counting from the beginning of each
burst: 0–3, 3–12, 12–18 and 18–36 s. The left part of Fig. 9 shows the light
curve of the source averaged over all X-ray bursts and the selected intervals
are highlighted by different colors. Results of applying the folding procedure
to the light curves extracted from these time intervals are shown on the right
panels of Fig. 9. In addition, we also provide the folded light curve of the
persistent emission of intervals of 1000 s before and after the bursts (the
bottom right panel). The pulse profiles practically do not change during the
burst and have a sinusoidal shape with a steeper left wing. This shape differs
significantly from the shape of the pulse profile of the persistent emission.
The fraction of pulsating emission in the 4–25 keV energy range remains
constant during the bursts, about 11%, and is similar to the value of the
pulsed fraction for persistent emission in the same energy band.

Figure 9: Pulse profile evolution during type-I X-ray bursts. Appropriate time
intervals are coded by color. The pulse profile of the persistent emission is
shown in black at the bottom right panel.

##  5 Discussion

We have reported the discovery of SRGA J1444, a new member of a small class of
AMXPs. We clearly detected pulsations from the source at a frequency of about
447.8 Hz and determined binary ephemeris based on the evolution of this
frequency over time. Our solution is in good agreement with the ephemerides
obtained from the data of NICER (Ray et al. 2024) and Insight-HXMT (Li et al.
2024) observatories. Based on the ephemeris, it is possible to obtain the
pulsar mass function $f_{\rm x}\simeq 0.00626\leavevmode\nobreak\ M_{\odot}$,
which, in turn, gives an estimate for the mass of a normal companion in this
system, $M_{2}>0.25\leavevmode\nobreak\ M_{\odot}$ (assuming the NS mass of
$M_{\rm NS}=1.4\leavevmode\nobreak\ M_{\odot}$).

###  5.1 Constraints from the burst/persistent X-ray emission

Based on the observational values, we can estimate some parameters of the
binary system or accretion flow. In particular, using the peak burst
luminosity value it is possible to estimate the distance to the system.
Indeed, if we take the empirical value of the Eddington luminosity
$(2.2-3.8)\times 10^{38}$ erg s-1 that depend only on the hydrogen abundance
in the burst fuel (Kuulkers et al. 2003) and the burst peak bolometric flux
$F_{\rm pk}=2.67\times 10^{-8}$ erg cm-2 s-1, then an upper limit on the
distance to the system will be $D<8.3-10.9$ kpc (a more advanced approach is
presented in the next section).

Figure 10: Dependence of the product ${\Delta tF_{4-25}^{\rm pst}}$ from time
for 17 pairs of consecutive bursts.

Our data provides an opportunity to evaluate another key parameter, $\alpha$\-
ratio of accretion to thermonuclear energy, which allows us to evaluate the
composition of bursting fuel, namely the mean hydrogen fraction at ignition,
$\overline{X}$. In other words, $\alpha$ shows how much the efficiency of
energy release during accretion process is higher than in a thermonuclear
reaction and can be calculated as a ratio of the energy released between two
consecutive bursts to the energy released per burst. In observational terms,
this parameter can be expressed as follows:

$\alpha=\frac{\Delta t\,F_{4-25}^{\rm pst}C_{\rm bol}}{E_{\rm b}},$ (1)

where $\Delta t$ is the burst recurrence time, $F_{4-25}^{\rm pst}$ is the
mean persistent flux between bursts, $C_{\rm bol}$ is the coefficient to
convert the persistent flux from a narrow energy band to bolometric flux and
$E_{\rm b}$ is the burst fluence (see Galloway et al. 2022, and references
there for more details). We have 17 pairs of consecutive bursts by which we
can estimate the product ${\Delta t\ F_{4-25}^{\rm pst}}$, and all products
within the error range take on the value $1.47\times 10^{-5}$ erg cm-2 (Fig.
10). We estimated $C_{\rm bol}$ using the model describing the spectrum of
persistent emission, extended the energy range to 0.2–60 keV (in which the
vast majority of energy is released) and according to the model the flux in
this energy range is about 3 times higher than the flux in the range 4–25 keV.
Bolometric fluence $E_{\rm b}=4.18\times 10^{-7}$ erg cm-2 we calculated based
on time resolved spectroscopy of the burst. Substituting the values obtained
above into Eq. (1) we get $\alpha\simeq 105$. From the measured value of
$\alpha$ we can estimate hydrogen mass fraction $\overline{X}$ in the ig
nition layer from Eq. (11) in Galloway et al. (2022). We get
$\overline{X}\approx 0.16$ assuming the NS radius $R_{\rm NS}=11.2$ km and
mass $M_{\rm NS}=1.4M_{\odot}$, indicating that the fuel is helium-rich.

###  5.2 X-ray bursts spectral evolution

Figure 11: Evolution of spectral parameters of the averaged X-ray burst from
SRGA J1444. The best-fit model curves for solar H/He mix (red dashed curve)
and pure He atmospheres (blue solid curve) are shown. The vertical dashed line
correspond to the Eddington flux for pure He atmosphere model.

Spectra of X-ray bursting NSs are well fitted by a diluted blackbody and can
be described by the model spectra of hot NS atmospheres (London et al. 1986;
Lewin et al. 1993; Suleimanov et al. 2012). Spectral evolution of X-ray bursts
which occurred during the hard persistent spectral state can be described with
the sequences of model atmosphere spectra of decreasing relative luminosities
(Suleimanov et al. 2012; Kajava et al. 2014). At the late burst stages, the
atmospheres of the X-ray bursting NSs can be heated by renewed accretion which
lead to certain changes in the emergent spectrum (Suleimanov et al. 2018).

Model hot NS atmosphere spectra can be approximated by a diluted blackbody
spectrum (see, e.g., Suleimanov et al. 2011)

$F_{E}=w\,\pi B_{E}(f_{\rm c}T_{\rm eff}),$ (2)

where the effective temperature $T_{\rm eff}$ is the parameter of the model
atmosphere, $w$ is the dilution factor, and $f_{\rm c}$ is the color
correction factor. Values of $w$ and $f_{\rm c}$ depend on $T_{\rm eff}$,
surface gravity, and chemical composition of the atmosphere. Extended tables
of $w$ and $f_{\rm c}$ for three chemical compositions (pure He, solar
abundance, and solar H/He mix with the metal abundances reduced by a factor of
100), nine surface gravities, and more than twenty relative luminosities, were
computed using the model atmospheres by Suleimanov et al. (2012). The
effective temperature for a given relative luminosity $\ell=L/L_{\rm Edd}$
depends on the surface gravity and the chemical composition. The tables of $w$
and $f_{\rm c}$ can be used for interpretation of the X-ray burst spectral
evolution after the maximum flux if the observed spectra are fitted by the
blackbody.

Figure 12: Same as Fig. 11, but showing the best-fits to the data with the
accretion-heated pure He atmosphere models of various accretion luminosities
$L_{\rm acc}/L_{\rm Edd}$=0 (solid line), 0.03 (dotted), and 0.05 (dashed)
(Suleimanov et al. 2017b). The vertical dashed line gives the Eddington flux
for the model with $L_{\rm acc}/L_{\rm Edd}$=0.03.

From the parameters of the blackbody fits shown in Fig. 4, we construct a
dependence of the blackbody normalization $K_{\rm bb}$ (in units (km/10 kpc)2)
on the bolometric flux $F_{\rm bb}$ (in erg s-1 cm-2) (see Figs. 11 and 12).
This observed dependence can be fitted with a model $(w\,f_{\rm
c}^{4})\,\ell-w$ using the direct cooling tail method (Suleimanov et al.
2017b). The theoretical models depend on the NS mass $M_{\rm NS}$ and radius
$R_{\rm NS}$, which therefore can be constrained for the given X-ray bursting
NS. The observed data for the X-ray bursts of the investigated source are not
good enough for using the direct cooling method. Therefore, we fixed the NS
mass $M_{\rm NS}=1.5\,M_{\odot}$ and considered the model curves computed for
the fixed surface gravity, $\log g=14.3$. In this case, we obtained two
fitting parameters, the observed Eddington flux $F_{\rm Edd}$ and the
geometrical dilution factor $\Omega$. They depend on the distance to the
source $D$, the NS radius $R_{\rm NS}$, and the gravitational redshift $z$ at
the NS surface:

$F_{\rm Edd}=\frac{GM_{\rm NS}\,c}{\kappa_{\rm
T}}\frac{1}{D^{2}\,(1+z)},\qquad\Omega=\frac{R_{\rm
NS}^{2}\,(1+z)^{2}}{D^{2}},$ (3)

where $\kappa_{\rm T}=0.2(1+X)$ cm2 g-1 is the electron scattering opacity and
$X$ is the hydrogen mass fraction in the atmosphere.

Table 1: Parameters of the NS atmosphere models fitted to the burst spectral
evolution.

Parameter | Solar | He (0.0) | He (0.03) | He (0.05)
---|---|---|---|---
$F_{\rm Edd}$ |  $\sim$4.28 |  $\sim$3.96 | $3.70^{+0.24}_{-0.21}$ | $3.27^{+0.28}_{-0.21}$
$\Omega$ (km/10 kpc)2 |  $\sim$275 |  $\sim$228 | $269^{+4}_{-2}$ | $305^{+10}_{-7}$
$D$ (kpc) |  $\sim$5.8 |  $\sim$8 |  $8.2\pm$1.0 | 8.8${}^{+1.1}_{-1.5}$
$R_{\rm NS}$ (km) |  $\sim$7.7 |  $\sim$9.6 |  $10.8\pm$1.1 | 12.3${}^{+1.0}_{-1.3}$

111NS mass was assumed to be $M_{\rm NS}=1.5\,M_{\odot}$ and gravity $\log
g$=14.3. The flux is measured in units of $10^{-8}$ erg s-1 cm-2. The best-fit
distance and the NS radius are obtained for the redshift $z=0.25$. The errors
correspond to the 68.3% confidence level. Numbers in brackets correspond to
the accretion luminosity in units of the Eddington one, $L_{\rm acc}/L_{\rm
Edd}$.

We first tried to describe the data by the models computed for two chemical
compositions, pure He and solar H/He mix. The results are presented in Fig. 11
and the first two columns of Table 1. We note that the obtained distances and
the NS radii are rather rough estimations only. However, we can conclude that
pure He composition of the NS atmosphere is preferred, because the radius
estimate is closer to the commonly accepted values of 11–13 km (Nättilä et al.
2016, 2017; Suleimanov et al. 2017b, a; Abbott et al. 2019; Riley et al.
2021). This conclusion is also in agreement with the low $X$ obtained on the
base of $\alpha$ value.

We use only two observational points in the presented above estimations.
Deviations of other points from the model curves can be explained by the
accretion heating (Suleimanov et al. 2018). Moreover, we can suggest that
accretion heats the NS atmosphere during the whole burst duration, because we
observe the pulsations during all burst phases (see Fig. 9). Therefore, we
tried to fit the observed data with the models computed for pure He, $\log
g=14.3$, and two accretion rates corresponding to $L_{\rm acc}/L_{\rm Edd}$ =
0.03 and 0.05. The results are presented in Fig. 12 and the two rightmost
columns of Table 1. These fits give even more acceptable NS radii of about
11–12 km. We note, however, that the cooling tail method is based on an
assumption that the NS surface emission is uniform. This is not correct for
the investigated source, because it shows pulsations during the bursts and
therefore the obtained estimates are approximate.

###  5.3 Modelling the pulse profiles

We found that the pulse profiles of the persistent emission of SRGA J1444 have
interesting shapes, following exactly the sine wave in the phase interval
0.0–0.5 and showing a plateau at phases 0.5–1.0 (Fig. 7). There is also some
energy dependence in the shape of the plateau. At high fluxes, a small bump
appears in the middle of the plateau (Fig. 8).

Figure 13: Normalized pulse profiles of the persistent emission of SRGA J1444
during the high state (upper data point at Fig. 7b shown with red crosses)
together with an example two-spot model (solid black curve). The dashed curve
gives the contribution from the primary spot.

The simplest explanation for the observed behavior is that two hotspots
(associated with the magnetic poles) contribute to the total emission. At
phases 0.0–0.5, only the primary spot, closest to the observer produces the
signal, while at phases 0.5–1.0 there is an additional contribution from the
secondary antipodal hotspot. Because of the large compactness two antipodal
spots can be seen together, and, if their emission pattern is close to the
blackbody, two sine waves coming out of phase cancel each other producing a
plateau (Beloborodov 2002; Poutanen & Beloborodov 2006).

However, the emission we see in the ART-XC energy range is clearly not a
blackbody, but produced by Comptonization in an optically thin region
(Poutanen & Gierliński 2003). Thus, the emission diagram is different and it
is not then obvious why the pulse profile has a plateau. Alternative
explanation involves the effects of partial eclipse of the secondary
(southern) hotspot by the accretion disk (Poutanen et al. 2009). For a stable
accretion to proceed, the inner radius of the accretion disk is expected to
lie within the corotation radius, which for a 447.8 Hz pulsar is $R_{\rm
co}=28.6\,(M_{\rm NS}/1.4\,M_{\odot})^{1/3}$ km. This is just a factor of 2.2
larger than the NS radius. Thus, the line of sight towards the southern
magnetic pole can pass through the accretion disk. The eclipse of the
secondary hotspot by the disk was likely responsible for a complex evolution
of the pulse profiles during the outburst of SAX J1808.4$-$3658 (Ibragimov &
Poutanen 2009; Poutanen et al. 2009; Kajava et al. 2011).

In order to model the pulse profiles, we consider two circular antipodal
hotspots at the NS surface, with the primary spot being at co-latitude
$\theta$. The angular size of the spot is $\rho$, the observer inclination
relative to the NS spin is $i$, and the inner radius of the accretion disk is
$R_{\rm in}$. The angular emission pattern of the intensity of radiation in
the spot frame is parametrized through a linear relation $I(\mu)\propto
1+h\cos\alpha^{\prime}$, where $\alpha^{\prime}$ is the angle relative to the
local normal in the spot comoving frame. Parameter $h$ takes the value of 0
for the blackbody emission, while a negative $h$ corresponds to the
Comptonized emission from an optically thin slab (Poutanen & Gierliński 2003;
Viironen & Poutanen 2004; Bobrikova et al. 2023). For simplicity, we consider
Schwarzschild metric and spherical shape of the NS surface. We used the code
described in Poutanen & Gierliński (2003) and Poutanen & Beloborodov (2006) to
model the light curves accounting for disk eclipse following formalism from
Ibragimov & Poutanen (2009). We fitted by eye the data corresponding to the
high state (Fig. 7b). We fixed NS mass to $1.4M_{\odot}$ and the radius to 12
km. We get a good description of the data for the pulsar inclination
$i=58\degr$, co-latitude of the spot center of 14° and its angular radius of
33°, the inner disk radius of 24.6 km, the anisotropy parameter $h=-0.63$, and
the phase shift of 0.43 (see Fig. 13).

As we noticed above, the pulse profiles during the bursts (Fig. 9) differ
substantially from those in the persistent emission. This is very different
from other bursting AMXPs, e.g., XTE J1814$-$338 and MAXI J1816$-$195
(Strohmayer et al. 2003; Ji et al. 2024), which show nearly identical
profiles. As a result, the same two-spot model with the disk eclipses does not
reproduce the data. Clearly some model modifications are required.

##  6 Summary

In this study, we present the results of the analysis of 133 ks of SRG/ART-XC
data, which led to the discovery of a new AMXP, SRGA J1444. Our main results
can be summarized as follows:

  * •

The timing analysis revealed a coherent signal near 447.8 Hz modulated by the
Doppler effect due to the orbital motion. The derived parameters for the
binary system are consistent with the circular orbit with a period of $\sim
5.2$ h.

  * •

The pulse profiles of the persistent emission, showing a sine-like part during
half a period with a plateau in between, can well be modelled by emission from
two circular spots partially eclipsed by the accretion disk.

  * •

The 19 thermonuclear X-ray bursts have been detected during observations. All
bursts have similar shapes and energetics and do not show any signs of
photospheric radius expansion. The burst rate decreases linearly from one per
$\sim$1.6 h at the beginning of observations to one per $\sim$2.2 h at the end
and anticorrelates with the persistent flux.

  * •

Spectral evolution during the bursts is consistent with the models of NS
atmospheres heated by accretion and implies a NS radius of 11–12 km and a
distance to the source of 8–9 kpc.

  * •

The pulsations during the bursts have been detected. We showed the pulse
profiles differ substantially from those observed in the persistent emission.
However, we could not find a simple physical model explaining the pulse
profiles detected during the bursts.

###### Acknowledgements.

This work is based on observations with the Mikhail Pavlinsky ART-XC
telescope, hard X-ray instrument on board the SRG observatory. The SRG
observatory was created by Roskosmos in the interests of the Russian Academy
of Sciences represented by its Space Research Institute (IKI) in the framework
of the Russian Federal Space Program, with the participation of Germany. The
ART-XC team thanks Lavochkin Association (NPOL) with partners for the creation
and operation of the SRG spacecraft (Navigator). This work was supported by
the grant Minobrnauki 23-075-67362-1-0409-000105. SST and JP acknowledge the
Academy of Finland grants 333112, 349373, and 349906. VFS thank the Deutsche
Forschungsgemeinschaft (DFG) for financial support (grant WE 1312/59-1).

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