# High-dimensional maximum-entropy phase space tomography using normalizing
flows

Austin Hoover<EMAIL_ADDRESS>Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37830, USA  Jonathan C. Wong  Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China

###### Abstract

Particle accelerators generate charged particle beams with tailored
distributions in six-dimensional (6D) position-momentum space (phase space).
Knowledge of the phase space distribution enables model-based beam
optimization and control. In the absence of direct measurements, the
distribution must be tomographically reconstructed from its projections. In
this paper, we highlight that such problems can be severely underdetermined
and that entropy maximization is the most conservative solution strategy. We
leverage normalizing flows—invertible generative models—to extend maximum-
entropy tomography to 6D phase space and perform numerical experiments to
validate the model performance. Our numerical experiments demonstrate that
flow-based entropy estimates are consistent with 2D maximum-entropy solutions
and that normalizing flows can fit complex 6D phase space distributions to
large measurement sets in reasonable time.

##  I Introduction

Accelerators generate charged particle beams with tailored distributions in
position-momentum space (phase space). Measuring the phase space distribution
enables model-based beam optimization and control, but such measurements are
challenging when the phase space is high-dimensional. If direct measurements
[1, 2, 3] are unavailable, the distribution must be tomographically
reconstructed from its projections. Fig. 1 shows a typical setup, where each
projection corresponds to the beam’s spatial distribution after transport
through an accelerator lattice (a series of electromagnetic focusing elements,
such as dipole or quadrupole magnets). In the most challenging versions of the
problem, we face a 4D or 6D reconstruction from 1D or 2D projections and
(potentially) nonlinear transformations [4, 5, 6, 7, 8, 9, 10, 11, 12].

Figure 1: Generic phase space tomography setup. The $n$-dimensional phase
space distribution $\rho(\bm{\mathbf{x}})$ travels through an accelerator
lattice—a series of electromagnetic focusing elements—represented by a
symplectic mapping $\bm{\mathbf{u}}_{k}=\mathcal{M}_{k}(\bm{\mathbf{x}})$. We
measure the $m$-dimensional projections $g(\bm{\mathbf{u}}_{k_{\parallel}})$
in the transformed space and use this data to reconstruct $\rho(\mathbf{x})$.
If the lattice contains nonlinear elements, the projections correspond to
curved integration surfaces in the original phase space.

It is possible for 6D reconstructions to be severely ill-posed. Since the
measured dimension is fixed, the set of feasible distributions—those
consistent with the measurements—may proliferate with the phase space
dimension. The total number of views can be limited to tens, compared to
hundreds in industrial or medical applications. We generally do not know the
information-maximizing set of 6D phase space transformations under given
measurement conditions, and accelerator constraints place many transformations
out of reach. And we do not know the level to which modeling or measurement
errors propagate through the 6D reconstruction.

Entropy maximization is a conservative approach to ill-posed inverse problems
and is thus well suited to high-dimensional tomography.111Entropy maximization
can be derived from self-consistency requirements on any strategy to infer
probability distributions from incomplete data [13, 14]. The maximum-entropy
distribution has minimal informational content, eliminating spurious
correlations and deviating from prior knowledge only to the extent required by
the data [15]. In this paper, we leverage normalizing flows—invertible
generative models—to extend maximum-entropy tomography to 6D phase space. Our
approach is a straightforward merger of two previous studies. Loaiza-Ganem,
Gao, and Cunningham [16], first proposed the use of flows for entropy
maximization subject to statistical moment constraints; we incorporate
projection constraints using the differentiable physics simulations and
projected density estimation proposed by Roussel et al. [11] in the Generative
Phase Space Reconstruction (GPSR) framework. We refer to the resulting
approach as MENT-Flow.

We begin by deriving the form of the $n$-dimensional maximum-entropy
distribution subject to $m$-dimensional projection constraints, extending the
2D and 4D analysis in [17, 7]. We then discuss the shortcomings of existing
maximum-entropy tomography algorithms when $n=6$ and describe the flow-based
solution. Finally, we perform numerical experiments to validate the model’s
reliability in 2D settings and examine the effects of entropic regularization
in 6D tomography.

##  II Maximum entropy tomography

Let $\rho(\bm{\mathbf{x}})$ be a probability distribution over phase space
coordinates $\bm{\mathbf{x}}\in\mathbb{R}^{n}$. We assume the $k$th
measurement occurs after a symplectic transformation
$\mathcal{M}_{k}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. Splitting the
transformed coordinates

$\bm{\mathbf{u}}_{k}=\mathcal{M}_{k}(\bm{\mathbf{x}})$ (1)

into a projection axis $\bm{\mathbf{u}}_{k_{\parallel}}\in\mathbb{R}^{m}$ and
orthogonal integration axis $\bm{\mathbf{u}}_{k_{\perp}}\in\mathbb{R}^{n-m}$,
we write the constraints as

$\displaystyle
g_{k}(\bm{\mathbf{u}}_{k_{\parallel}})-\int\rho\left(\mathcal{M}_{k}^{-1}(\bm{\mathbf{u}}_{k})\right)d\bm{\mathbf{u}}_{k_{\perp}}=0$
(2)

for measured projection $g_{k}(\bm{\mathbf{u}}_{k_{\parallel}})$. Letting
prior $\rho_{*}(\bm{\mathbf{x}})$ encode our knowledge of the distribution in
the absence of data, we aim to maximize the entropy

$H[\rho(\bm{\mathbf{x}}),\rho_{*}(\bm{\mathbf{x}})]=-\int{\rho(\bm{\mathbf{x}})\log{\left(\frac{\rho(\bm{\mathbf{x}})}{\rho_{*}(\bm{\mathbf{x}})}\right)}d\bm{\mathbf{x}},}$
(3)

subject to Eq. (2). We will refer to this problem as an $n$:$m$
reconstruction.

We can derive the form of the maximum-entropy distribution from a new
functional

$\Psi[\rho(\bm{\mathbf{x}}),\rho_{*}(\bm{\mathbf{x}}),\\{g_{k}(\bm{\mathbf{u}}_{k_{\parallel}})\\}]=H[\rho(\bm{\mathbf{x}}),\rho_{*}(\bm{\mathbf{x}})]+\sum_{k}{\int\lambda_{k}(\bm{\mathbf{u}}_{k_{\parallel}})\left(g_{k}(\bm{\mathbf{u}}_{k_{\parallel}})-\int\rho\left(\mathcal{M}_{k}^{-1}(\bm{\mathbf{u}}_{k})\right)d\bm{\mathbf{u}}_{k_{\perp}}\right)d\bm{\mathbf{u}}_{k_{\parallel}}},$
(4)

where $\lambda_{k}(\bm{\mathbf{u}}_{k_{\parallel}})$ are unknown functions
[18]. Enforcing zero variation of $\Psi$ with respect to
$\rho(\bm{\mathbf{x}})$ and $\lambda_{k}(\bm{\mathbf{u}}_{k_{\parallel}})$
gives

$\displaystyle\rho(\bm{\mathbf{x}})$
$\displaystyle=\rho_{*}(\bm{\mathbf{x}})\prod_{k}\exp{\left(\lambda_{k}(\bm{\mathbf{u}}_{k_{\parallel}}(\bm{\mathbf{x}}))\right)}$
(5)
$\displaystyle=\rho_{*}(\bm{\mathbf{x}})\prod_{k}h_{k}(\bm{\mathbf{u}}_{k_{\parallel}}(\bm{\mathbf{x}})).$

The constrained maximum-entropy distribution function at point
$\bm{\mathbf{x}}$ is the product of two terms: the prior distribution function
at $\bm{\mathbf{x}}$ and the product of the positive Lagrange functions
$h_{k}$ evaluated at the corresponding points
$\bm{\mathbf{u}}_{k_{\parallel}}(\bm{\mathbf{x}})$ on the measurement
axes.222In MENT literature, the Lagrange multipliers/functions often refer to
$\lambda_{k}$ in Eq. (5), but here we refer to the exponential of
$\lambda_{k}$. Substituting Eq. (5) into Eq. (2) generates a set of coupled
nonlinear integral equations from which $h_{k}$ are to be solved.

###  II.1 MENT

The MENT algorithm [17, 19, 18, 7] leverages a Gauss-Seidel relaxation method
to optimize the Lagrange functions in Eq. (5). After initializing the
distribution to the prior within the measurement boundaries:

$h_{k}(\bm{\mathbf{u}}_{k_{\parallel}})=\begin{cases}1,&\text{if}\
g_{k}(\bm{\mathbf{u}}_{k_{\parallel}})>0\\\ 0,&\text{otherwise}\end{cases}.$
(6)

the Lagrange functions are updated as

$h_{k}(\bm{\mathbf{u}}_{k_{\parallel}})\leftarrow
h_{k}(\bm{\mathbf{u}}_{k_{\parallel}})\left(1+\omega\left(\frac{g_{k}(\bm{\mathbf{u}}_{k_{\parallel}})}{\tilde{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}})}-1\right)\right)$
(7)

where

$\displaystyle\tilde{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}})$
$\displaystyle=\int{\rho_{*}\left(\mathcal{M}_{k}^{-1}(\bm{\mathbf{u}}_{k})\right)\prod\limits_{j}h_{j}(\bm{\mathbf{u}}_{j_{\parallel}})d\bm{\mathbf{u}}_{k_{\perp}}}$
(8)
$\displaystyle=h_{k}(\bm{\mathbf{u}}_{k_{\parallel}})\int{\rho_{*}\left(\mathcal{M}_{k}^{-1}(\bm{\mathbf{u}}_{k})\right)\prod\limits_{j\neq
k}h_{j}(\bm{\mathbf{u}}_{j_{\parallel}})d\bm{\mathbf{u}}_{k_{\perp}}}$

are the simulated projections and $0<\omega\leq 1$ is a learning rate [18].
The updates are performed in order ($k=1,2,3,\dots$), and each updated $h_{k}$
is immediately used to simulate the next projection. One epoch is completed
when all functions are updated. The iterations in Eq. (7) converge [17, 19].

MENT is a powerful tomographic algorithm. Its unique advantage is that it
maximizes entropy by design: fitting the data generates an exact solution to
the constrained optimization problem. MENT is also efficient (it stores the
exact number of parameters needed to define the maximum-entropy distribution
and typically converges in a few epochs) and essentially hyperparameter-free.
Additionally, although not yet reported in the literature, MENT does not
assume the phase space transformations are linear.

We showed above that MENT extends to $n$:$m$ tomography. In practice, the
value of $n$ is limited by the integrals required to compute the
distribution’s projections. Ongoing work aims to demonstrate efficient
implementations when $n=4$ [7, 20]. Extension to $n=6$ may be possible, but it
has not been demonstrated, and the runtime may be slow if there are many high-
resolution measurements. Additionally, even if the algorithm converged,
generating particles would require sampling from an expensive-to-evaluate 6D
distribution function.

###  II.2 MENT-Flow

In the absence of a method to directly optimize the Lagrange functions in Eq.
(5), we may try to minimize the loss function

$L[\rho(\bm{\mathbf{x}}),\rho_{*}(\bm{\mathbf{x}}),\\{{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}})\\}]=-H[\rho(\bm{\mathbf{x}}),\rho_{*}(\bm{\mathbf{x}})]+\mu\sum_{k}{D[{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}}),\tilde{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}})]}$
(9)

for an increasing sequence of penalty parameters $\mu$. Here,
$D[{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}}),\tilde{g}_{k}(\bm{\mathbf{u}}_{k_{\parallel}})]$
is a non-negative number quantifying the discrepancy between the measured and
simulated projections, which we choose to be the Kullback-Leibler (KL)
divergence. Although exact solutions may require $\mu\rightarrow\infty$,
approximate solutions obtained with finite $\mu$ are often sufficient.333More
sophisticated Augmented Lagrangian (AL) methods [16, 21] are available to
stabilize the optimization and avoid large penalty parameters, but we did not
see significant improvements using AL in our numerical experiments. This
approach requires a parameterization of $\rho(\bm{\mathbf{x}})$. Direct
parameterizations require 6D integration or Monte Carlo sampling to compute
the distribution’s entropy and projections. Indirect parameterizations instead
define

$\bm{\mathbf{x}}=\mathcal{F}(\bm{\mathbf{z}}),$ (10)

where $\mathbf{z}$ is drawn from a base distribution
$\rho_{0}(\bm{\mathbf{z}})$ defined in a normalized space. Typically,
$\rho_{0}(\bm{\mathbf{z}})$ is an easy-to-sample-from distribution such as a
standard Gaussian distribution. Roussel et al. [11] adopted this approach,
using a neural network to represent $\mathcal{F}$ and using differentiable
beam dynamics simulations and kernel density estimation to fit the network
parameters to projection data. However, it is difficult to maximize the
entropy without access to $\rho(\bm{\mathbf{x}})$.444Roussel et al. [11]
proposed to maximize the emittance, or root-mean-square (rms) volume,
$\varepsilon={|\bm{\mathbf{\Sigma}}|}^{1/2}$, where
$\bm{\mathbf{\Sigma}}=\langle\bm{\mathbf{x}}\bm{\mathbf{x}}^{T}\rangle$ is the
$n\times n$ covariance matrix of second order moments, as a proxy for the
entropy. For certain distributions, the logarithm of the emittance is
proportional to the entropy, but this is not true in general. Like entropy
maximization, emittance maximization removes unnecessary linear correlations
from the reconstructed distribution. However, it cannot remove nonlinear
correlations, as the emittance depends only on second-order moments.
Furthermore, the maximum-emittance distribution is not unique. In linear
systems, the covariance matrix is typically overdetermined by the tomographic
measurements, i.e., all distributions that fit the data have the same
emittance. Particle-based entropy estimates based on k nearest neighbors [22]
may perform better.

Normalizing flows are indirect parameterizations that provide access to the
probability density. Flows define
$\mathcal{F}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ as an invertible,
differentiable map and compute the density from

$\log{\rho(\bm{\mathbf{x}})}=\log{\rho_{0}(\bm{\mathbf{z}})}-\log{\left|{\det
J_{\mathcal{F}}(\bm{\mathbf{z}})}\right|},$ (11)

where

$J_{\mathcal{F}}(\bm{\mathbf{z}})=\frac{d\mathcal{F}}{d\bm{\mathbf{z}}}=\begin{bmatrix}\frac{\partial{\mathcal{F}_{1}}}{\partial{z_{1}}}&\dots&\frac{\partial{\mathcal{F}_{1}}}{\partial{z_{n}}}\\\
\vdots&\ddots&\vdots\\\
\frac{\partial{\mathcal{F}_{n}}}{\partial{z_{1}}}&\dots&\frac{\partial{\mathcal{F}_{n}}}{\partial{z_{n}}}\\\
\end{bmatrix}.$ (12)

is the $n\times n$ Jacobian matrix of $\mathcal{F}$. Given $N$ flow samples
$\\{\bm{\mathbf{x}}_{i}\\}$, an unbiased entropy estimate is available: [16]:

$\displaystyle-H[\rho(\bm{\mathbf{x}}),\rho_{*}(\bm{\mathbf{x}})]$
$\displaystyle=\mathbb{E}_{\rho(\bm{\mathbf{x}})}\left[\log\rho(\bm{\mathbf{x}})-\log\rho_{*}(\bm{\mathbf{x}})\right]$
(13)
$\displaystyle\approx\frac{1}{N}\sum_{i=1}^{N}\left(\log\rho(\bm{\mathbf{x}}_{i})-\log\rho_{*}(\bm{\mathbf{x}}_{i})\right)$

Since $\mathcal{F}$ is differentiable, we can maximize the entropy estimate
via, e.g., stochastic gradient descent [16]. We call this approach MENT-Flow
when combined with differentiable tomographic constraints.

It is not immediately obvious whether flows can learn complex 6D phase space
distributions from projection data in reasonable time. Flows are restricted to
invertible transformations that preserve the topological features of the base
distribution; for example, flows cannot perfectly represent disconnected modes
if the base distribution has a single mode [23]. Building complex flows
requires layering transformations, either as a series of maps (discrete flows)
or a system of differential equations (continuous flows), often leading to
large models and expensive training.555A relevant example comes from Green,
Ting, and Kamdar [24], who used continuous flows for 6D phase space density
estimation from measured stellar phase space coordinates; training times
ranged from hours to days on a GPU, depending on the distribution complexity,
with approximately $10^{4}$ particles per batch.

We found that Neural Spline Flows (NSF) [25] provide a sufficient blend of
speed and power. Each flow layer is an autoregressive transformation of the
form

$z_{i}\rightarrow\mathcal{F}(z_{i};c_{i}(z_{1},\dots,z_{i-1})),$ (14)

where $\bm{\mathbf{z}}=[z_{1},\dots,z_{n}]^{T}$. Here, $\mathcal{F}$ is an
invertible function conditioned on $c_{i}(z_{1},\dots,z_{i-1})$. The
conditioner generates a triangular Jacobian whose determinant can be
efficiently computed. We implemented the conditioner using a masked neural
network [26] and the transformer using rational-quadratic splines [25].
Representational power increases with the number of neural network parameters,
the number of knots in the spline transformer, and the number of flow layers.
For the composition of $T$ layers

$\mathcal{F}=\mathcal{F}_{T}\circ\mathcal{F}_{T-1}\circ\dots\circ\mathcal{F}_{2}\circ\mathcal{F}_{1},$
(15)

and transformed coordinates

$\bm{\mathbf{z}}_{t}=\mathcal{F}_{t}(\bm{\mathbf{z}}_{t-1}),$ (16)

the Jacobian determinant is available from

$\left|J_{\mathcal{F}}(\bm{\mathbf{z}}_{0})\right|=\prod_{t=1}^{T}{\left|J_{\mathcal{F}_{t}}(\bm{\mathbf{z}}_{t-1})\right|.}$
(17)

MENT-Flow increases the reconstruction model complexity and does not guarantee
an exact entropy maximum. However, MENT-Flow scales straightforwardly to
$n$-dimensional phase space and immediately generates independent and
identically distributed samples from the reconstructed distribution.

##  III Numerical experiments

The following numerical experiments demonstrate that MENT-Flow solutions
approach MENT solutions in 2D phase space. Subsequent experiments demonstrate
that MENT-Flow can fit complicated 6D phase space distributions to large
measurement sets in reasonable time and that entropic regularization keeps the
reconstruction close to the prior distribution. We tended to use ground-truth
distributions without linear interplane correlations and simple
transformations rather than realistic accelerator models.666Equivalently, we
assume we know the covariance matrix
$\bm{\mathbf{\Sigma}}=\langle\bm{\mathbf{x}}\bm{\mathbf{x}}^{T}\rangle=\bm{\mathbf{V}}\bm{\mathbf{V}}^{T}$
and reconstruct the distribution in normalized coordinates
$\bm{\mathbf{x}}_{n}=\bm{\mathbf{V}}^{-1}\bm{\mathbf{x}}$ by setting
$\mathcal{M}_{k}\rightarrow\mathcal{M}_{k}\bm{\mathbf{V}}$. The covariance
matrix is usually overdetermined by the measurements—for example, three
measurements determine the $2\times 2$ covariance matrix—so that all
distributions that fit the data share the same covariance matrix. In these
cases it is reasonable to fit the covariance matrix first. We also chose to
maximize the entropy relative to a Gaussian prior distribution
($\rho_{*}(\bm{\mathbf{x}})$) and used a Gaussian base distribution, making
the maximum-entropy flow a linear transformation.777A Gaussian prior may be a
reasonable choice for accelerator applications: the prior has no interplane
dependence and can expand to approximate a uniform distribution; (ii) any
known elements of the $n\times n$ covariance matrix can be used to define the
Gaussian prior; (iii) beams are typically clustered in phase space and
approximately Gaussian at equilibrium; (iv) a Gaussian prior can be used to
limit the beam size in dimensions that are weakly constrained by the data; and
(v) the relative entropy has a maximum of zero. All training and analysis code
is available in an online repository [27].

###  III.1 2D reconstructions from 1D projections

Our first experiment tests the model performance in 2:1 phase space
tomography. We assume a linear accelerator lattice, composed of drifts and
quadrupole magnets, such that a symplectic transfer matrix $\bm{\mathbf{M}}$
approximates the dynamics. The transfer matrix can be decomposed as

$\bm{\mathbf{M}}=\bm{\mathbf{V}}(\alpha_{2},\beta_{2})\bm{\mathbf{R}}(\mu)\bm{\mathbf{V}}(\alpha_{1},\beta_{1})^{-1},$
(18)

where

$\bm{\mathbf{V}}(\alpha,\beta)=\begin{bmatrix}\sqrt{\beta}&0\\\
-\frac{\alpha}{\sqrt{\beta}}&\frac{1}{\sqrt{\beta}}\end{bmatrix}$ (19)

is a normalization matrix, parameterized by $\alpha$ and $\beta$, and

$\bm{\mathbf{R}}(\mu)=\begin{bmatrix}\cos\mu&\sin\mu\\\
\sin\mu&\cos\mu\end{bmatrix}$ (20)

is a rotation by the phase advance $\mu$. The projection angle, and hence the
reconstruction quality, depends only on the phase advance. Various constraints
can limit the projection angle range, but we assume the projection angles are
evenly spaced over the maximum 180-degree range.

Fig. 2 shows reconstructions from a varying number of projections, comparing
MENT, MENT-Flow, and an unregularized neural network (NN).

Figure 2: 2D reconstructions from varying numbers of evenly spaced 1D
projections. The top rows plot samples from the true distribution, MENT
reconstruction, MENT-Flow reconstruction, and unregularized neural network
reconstruction. Faint lines show the evenly spaced projection angles,
increasing from 1 on the left column to 7 on the right column. The projections
of the four distributions are plotted on the horizontal axes on the bottom
rows.

It is clear that maximizing the stochastic estimate in Eq. (13) pushes the
distribution’s entropy close to its constrained maximum. (Recall that MENT
maximizes entropy by construction). Although the MENT solutions are of higher
quality, the differences are not visible from afar.

Fig. 2 illustrates that entropy maximization is a conservative approach to the
reconstruction problem. All reconstructed features are implied by the data. In
contrast, the distributions in the bottom rows fit the data but are
unnecessarily complex. Of course, reconstructions from one or two projections
are bound to fail, no matter the algorithm, but these cases are still useful
because they demonstrate MENT’s logical consistency: given only the marginal
distributions and an uncorrelated prior distribution, MENT returns the product
of the marginals. On the other extreme, with enough data, the feasible
distributions will differ only in minor details. MENT shines in intermediate
cases where the measurements contain just enough information to constrain the
distribution’s primary features. For example, the continuous spiral structure
develops rapidly with the number of views.

Fig. 2 also illustrates the flow’s capacity to represent complex distributions
despite its restriction to invertible transformations. This examples focuses
on spiral patterns, which are characteristic of nonlinear dynamics.
(Additional examples are included in the supplemental material.) It is
important to note that, while our analysis focuses on the beam core, low-
density regions can also impact accelerator performance [28]. Flows can
struggle to model distribution tails when trained via maximum-likelihood [29].
Our example distribution does not have significant halo and we do not report
the agreement at this level, but preliminary studies indicate that the
Kullback-Leibler (KL) divergence may enhance dynamic range relative to, i.e.,
the mean absolute error when fitting the data.

###  III.2 6D reconstructions from 1D projections

It is more challenging to design and evaluate 6D numerical experiments. First,
we do not know the information-maximizing set of 6D phase space
transformations. In the 2D case, if the transformations are linear, there is a
clear connection between a single transformation parameter (the phase advance)
and the reconstruction quality. 4D or 6D symplectic transformations require
many more parameters [30], and it is unclear which parameters are relevant.
Note also that the ideal set of transformations depends on the distribution
and on the number of measured dimensions. Here, we restrict our attention to
1D projections, i.e., 6:1 reconstructions, for which we can plausibly obtain
the ideal set of linear phase space transformations by generalizing the notion
of“projection angle”. A 1D projection axis is defined by a point on the unit
$n$-sphere; if the distribution is spherically symmetric, we expect the
optimal projection axes to be uniformly distributed on the sphere.888It is
unclear what “evenly spaced projection angles” correspond to when the
integration planes have dimension $n-2$. In the limit of many such
projections, the reconstruction will converge to the true distribution [31].

Second, there are many possible 6D distributions, and establishing
reconstruction accuracy requires high-dimensional visualization or statistical
distance metrics. As described shortly, we selected distributions with clear
high-dimensional structure and leveraged complete sets of pairwise projections
and limited sets of partial projections, i.e., projections of slices, to
facilitate the visualization task.

Third, we cannot determine the distance from the reconstructed distribution to
the true maximum-entropy distribution without an analytic solution. We point
to Fig. 2 as evidence that the flow’s entropy estimate can approximate the
true entropy, and we continue to train an unregularized neural network on the
same data to show that additional solutions can exist far from our prior.

Our first high-dimensional experiment, shown in Figs. 3-4, reconstructs a
seven-mode Gaussian mixture distribution from 25, then 100 random 1D
projections. This reconstruction used an approximately uniform prior and the
same flow architecture as our 2D experiment.

(a)

(b)

(c)

(d)

Figure 3: Reconstruction of 6D Gaussian mixture distribution from 25 random 1D
projections. (a-b) Simulated/true projections. (c-d) 1D/2D projections of
reconstructed 6D distribution. Reconstructed/true 2D projections are plotted
on the lower/upper triangular; 1D projections are overlayed on the diagonal.

(a)

(b)

(c)

(d)

Figure 4: Reconstruction of 6D Gaussian mixture distribution from 100 random
1D projections. (a-b) Simulated/true projections. (c-d) 1D/2D projections of
reconstructed 6D distribution. Reconstructed/true 2D projections are plotted
on the lower/upper triangular; 1D projections are overlayed on the diagonal.

We draw the following conclusions. (i) Flow-based models can represent complex
6D densities far from their unimodal base densities. All simulated
measurements agree with the ground truth. Charged particle beams are often
smooth and unimodal; therefore, we presume that flow-based models will be
sufficient for most applications in accelerator physics. (ii) Some complex,
high-dimensional distributions require large numbers of measurements to
reconstruct with high precision. Of course, the required precision varies with
application. In our case, the distribution’s primary features are largely
determined from only 25 measurements in Fig. 3, but the entropy penalty
eliminates unnecessary high-frequency terms in the distribution function.
These terms are still present in the unregularized reconstruction when we
quadruple the number of measurements in Fig. 4. (iii) MENT-Flow can scale to
large measurement sets in reasonable time.

The Gaussian mixture distribution has little overlap between modes, so
mismatch between the true and reconstructed distribution is obvious from low-
dimensional views. This is not always the case. Direct measurements of low-
energy hadron beams have unveiled hollow structures in high-dimensional phase
space which are not evident in low-dimensional views.999Measurements at the
Spallation Neutron Source (SNS) Beam Test Facility (BTF) [1] show space-
charge-driven hollowing in 3D and 5D projections of the 6D phase space
distribution. This motivates us to test tomographic reconstruction algorithms
on distributions with hidden internal structure. To this end, an
$n$-dimensional “rings” distribution serves as the ground truth in Fig. 5-6;
particles populate two concentric $n$-spheres with radii $r_{2}=2r_{1}$, and
the radii are perturbed with Gaussian noise to generate a smooth density.

(a)

(b)

(c)

(d)

Figure 5: Reconstruction of 6D “rings” distribution from 25 random 1D
projections. (a-b) Simulated/true projections. (c-d) Reconstructed 2D
projections within shrinking 4D ball of radius
$\tilde{r}_{\perp}=\sqrt{x^{2}+x^{\prime 2}+y^{2}+y^{\prime 2}}$ in unplotted
dimensions.

(a)

(b)

(c)

(d)

Figure 6: Reconstruction of 6D “rings” distribution from 100 random 1D
projections. (a-b) Simulated/true projections. (c-d) Reconstructed 2D
projections within shrinking 4D ball of radius
$\tilde{r}_{\perp}=\sqrt{x^{2}+x^{\prime 2}+y^{2}+y^{\prime 2}}$ in unplotted
dimensions.

The entropy-regularized distribution maintains the spherical symmetry of the
Gaussian prior, flattening and eventually inverting its radial density profile
to fit the data.101010The one-dimensional projections in Figs. 5-6 are nearly
Gaussian. Klartag [32] proved that almost all $m$-dimensional projections of
an isotropic (no linear correlations) $n$-dimensional log-concave distribution
function are nearly Gaussian when $n\gg m$. Many distributions commonly used
in accelerator modeling are log-concave, such as the $n$-dimensional Gaussian,
Waterbag (uniformly filled ball), and KV (uniformly filled sphere)
distributions. A practical implication of this theorem is that small
fluctuations in the $m$-dimensional projections have a greater impact on the
$n$-dimensional reconstructed distribution as $|n-m|$ increases—for instance,
completely inverting the density profile from peaked to hollow. Thus, we found
that later training epochs can significantly change the distribution while
only slightly decreasing the loss function. It follows that, for certain
distributions, there may be some value of $|n-m|$ for which $n$:$m$ tomography
is practically impossible. The sliced views reveal an internal structure—a
dense core surrounded by a low-density cloud—that is better approximated by
MENT-Flow when measurements are scarce. In addition to injecting unnecessary
correlations between planes, the unregularized solution ejects all particles
from the core. Surprisingly, adding additional measurements does not solve the
problem and generates two distinct modes in the reconstructed density. Using a
different random seed to define the measurement axes can generate different
patterns, but the hollowing and splitting just described are typical. Note
that this internal structure is not obvious from the full 2D projections in
the left column of Figs. 5-6. In experimental settings, we often have strong a
priori assumptions of smoothness and unimodality based on beam dynamics
simulations and theory. By encoding this background knowledge in MENT-Flow’s
prior distribution, we are likely to obtain more reliable high-dimensional
reconstructions.

##  IV Conclusion and extensions

MENT-Flow is a promising approach to high-dimensional maximum-entropy phase
space tomography. Numerical experiments demonstrate consistency with known 2D
maximum-entropy solutions and the ability to fit complex 6D phase space
distributions to large measurement sets. Our numerical experiments also
emphasize the importance of regularization and uncertainty quantification in
high-dimensional phase space tomography. We demonstrated that large numbers of
1D projections are insufficient to reconstruct certain complex 6D
distributions with high precision. Future work should apply the method to more
realistic distributions and phase space transformations. Additionally, future
work should study the importance of entropy regularization when the
measurements are 2D projections (which we expect to contain exponentially more
information per measurement) or come from alternative diagnostics [33]. At the
same time, alternative maximum-entropy algorithms should be investigated to
address a wider range of problems.

MENT-Flow’s primary shortcomings are its slow sampling speed, its large number
of parameters, and its restriction to differentiable particle dynamics.
Sampling is approximately 50% slower than a conventional neural network, and
the total runtime is increased by the need to solve multiple subproblems to
approach the maximum-entropy distribution from below. This motivates the
search for even more efficient flows and sample-based entropy estimates. Note,
however, that the total training time is already reasonable, ranging from 3 to
15 minutes on a single GPU in our experiments (depending on the number of
projections, phase space dimension, batch size, and penalty parameter
updates).

Flow-based phase space tomography may be able to approach maximum-entropy
solutions to unexplored problems. First, MENT-Flow may be able to reconstruct
$n$-dimensional distributions from high-dimensional projections ($m>2$). Such
problems are of theoretical interest but also have some practical relevance.
3D and 4D projections can be measured relatively quickly using slit-screen-
dipole measurement systems in low-energy hadron accelerators [1, 2, 3].
Projections may be partial, i.e., defined within a slice along other
dimensions. The differentiable kernel density estimation needed to simulate
the projections will perform poorly in these cases, so we propose to draw
samples from the measured projections and minimize a statistical distance
between these samples and samples from the flow.

Next, a particularly interesting application of maximum-entropy tomography is
to intense hadron beams. Since space charge can generate highly nonlinear
self-fields that depend on the 3D spatial distribution, knowledge of the full
6D phase space distribution is needed to predict the evolution of intense
beams. A common measurement setup is to place the reconstruction location
upstream of the measurement location, thereby allowing low-dimensional
measurements under varying transport conditions (e.g. quadrupole scan) to
obtain more information about the beam at the reconstruction point.
Introducing transport renders the tomography problem a significant challenge
for intense beams because the forward process depends on self-fields of the
unknown phase space distribution. With MENT-Flow, it may be possible to
include space charge in the beam dynamics using differentiable space charge
models [34]. We also plan to pursue tomography under intense space charge with
conventional MENT, which may provide a benchmark for MENT-Flow results.

Finally, quantifying reconstruction uncertainty will be a crucial step for the
operational use of high-dimensional phase space tomography. Of particular
interest are Bayesian methods, where a prior probability is assigned to every
possible phase space distribution and updated after observing the data. (The
principle of maximum entropy claims that the prior should increase as a
function of entropy [13], so the strategy in this paper remains correct if we
must select a single distribution.) Several authors have used generative
models to solve inverse image reconstruction problems via Bayesian posterior
sampling [35]. Perhaps similar methods could be applied to our case.

##  V Acknowledgements

We are grateful to Ryan Roussel (SLAC National Accelerator Laboratory), Juan
Pablo Gonzalez-Aguilera (University of Chicago), and Auralee Edelen (SLAC
National Accelerator Laboratory) for discussions that seeded the idea for this
work and for sharing their differentiable kernel density estimation code.

This manuscript has been authored by UT Battelle, LLC under Contract No. DE-
AC05-00OR22725 with the U.S. Department of Energy. The United States
Government retains and the publisher, by accepting the article for
publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes. The Department of Energy will provide public
access to these results of federally sponsored research in accordance with the
DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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has been achieved by capacitive coupling with a transmission line resonator
with characteristic resonance $\omega_{c}/(2\pi)\sim 7.2$GHz [75, 78, 70].
Two-tone spectroscopy as a function of offset charge $n_{g}$ and external flux
$\Phi_{{\rm ext}}$ has been employed to determine the spectrum of the circuit,
finding a remarkable agreement with the simple two-mode Hamiltonian
$H_{\theta\phi}$ in Equation (40).

Direct transitions between the computational states are strongly suppressed,
because of their disjoint support. The control of the qubit has been therefore
achieved by taking advantage of higher-energy states, as the latter have
support on both the $\theta=0$ and $\theta=\pi$ valleys [see Figure 5(e)].
While this is reminiscent of the protocol proposed in Ref [137] and mentioned
in Section 3.5, these two schemes are actually based on different strategies.
In particular, the experimental implementation relied on Raman processes that
virtually populate one ancillary excited state. This has the advantage that
only a very small fraction of the population occupies the unprotected
intermediate state but, unfortunately, its applicability is limited to the
soft regime since, in the deep one, the onset of an approximate selection rule
makes Raman-based gates ineffective [137]. This limitation is overcome by the
deep regime-ready protocol described in Section 3.5, where several excited
states are actually populated for a very short time. Nevertheless, for current
experimental realizations of the $0-\pi$ qubit, a Raman gate scheme is
effective in controlling population transfer between the two computational
states, as demonstrated by clear Rabi oscillations reported in Ref. [70].
There, an impressive relaxation decay time $T_{1}\sim 1.6$ms has been also
recorded, as shown in Figure 10(b), which is larger than state-of-the-art
transmon performance [119, 120], and comparable to the last achievement in
fluxonium device [80]. As for the overall coherence times, obtained by Ramsey
interferometry and Hahn echo protocol, values of $T_{2}^{*}\sim 9\ \mu$s and
$T_{2e}\sim 25\ \mu$s have been extracted, demonstrating first order
protection from flux noise [70]. This first experimental realization of
$0-\pi$ circuit confirms the huge potential of these protected superconducting
circuits, opening new perspectives for the implementation of protected multi-
qubits gates and more complex operations.

###  4.5 Kinetic interference cotunneling element

We now describe the implementation of a generalized Josephson element [173],
first proposed in Ref [65], called Kinetic Interference co-Tunneling Element
(KITE). Also this superconducting circuit is able to enter a protected regime,
where the transport is dominated by co-tunneling events of pair of Cooper
pairs. Furthermore, by investigating its flux response, it is possible to find
a direct link between protected regime and an enhancement of zero-point phase
fluctuations. This fact therefore reduces the asymmetry between charge and
flux fluctuations discussed in Section 2.3.2, paving the way for artificial
systems where the two can act in a symmetric fashion.

The circuit consists of a superconducting loop where a parallel of two small
identical JJs (with Josephson energy $E_{J}$ and charging energy $E_{C}$) is
placed in series with two superinductances (with associated energy $E_{L}$),
see the green portion of the sketch in Figure 11(a). The parameter regime
implemented here is $E_{L}<E_{CL}\ll E_{J}\sim E_{C}$ with $E_{L}=0.23$GHz,
$E_{CL}=2.5$GHz, $E_{J}=5.9$GHz, and $E_{C}=6.6$GHz. This configuration, in
contrast with the rhombus element [58, 66, 67], does not contain any central
superconducting island, and it is thus less sensitive to offset charge noise.

Figure 11: (a) Schematic of the KITE device. (b,c) Two-tone spectroscopy
measurements of the transition frequencies from the ground state, for
$\theta_{\rm ext}=0$ (b) and $\theta_{\rm ext}=\pi$, respectively. All panels
adapted from Ref. [173]

Thanks to the presence of an additional shunting large inductance (see blue
region in the figure), this device can pass from a conventional $2\pi$ to a
$\pi$ periodicity, depending on the external flux threading the whole circuit.
The latter behavior occurs at half flux and it is caused by Aharonov-Bohm
destructive interference of single Cooper pair tunneling, exactly as in the
case of a rhombus element [62, 151]. We notice that in the experimental
realization both superinductors have been realized by JJ arrays containing
$100$ single large JJ elements by standard double angle Al evaporation
technique. The circuit has three degrees of freedom, two associated with the
KITE and one with the shunting portion. Since $E_{C}\sim E_{J}$ is the
dominant energy scale, the system can be effectively described in terms of
only two phases by relying on the Born-Oppenheimer approximation. Tuning the
external flux from 0 to half flux $\theta_{{\rm ext}}=\pi$, the reduced
Hamiltonian can be further simplified with the following one-dimensional
effective description

$H_{\mu}=4E_{C}\left(\frac{n}{\mu}\right)^{2}+\frac{E_{L}}{2}(\varphi-\varphi_{{\rm
ext}})^{2}+(-1)^{\mu}E_{J}\cos(\mu\varphi)~{},$ (74)

where $\varphi_{{\rm ext}}$ is associated to the external flux across the
suprconducting loop and the parameter $\mu=1,2$ determine the two possible
case, respectively. Note that the $(-1)^{\mu}$ factor guarantees that the
cosine potential is pinned to 1 at $\pm\pi$ for all $\mu=1,2$. Device
properties have been measured with an inductively coupled LC read-out
resonator. Circuit transition frequencies have been probed by two-tone
spectroscopy. It has been demonstrated [173] the halving of periodicity, from
conventional $2\pi$ at zero flux to $\pi$ at half flux.

Correspondingly, when the period is halved the ground state wavefunction
delocalization gets magnified. This is confirmed by the suppression of flux
dispersion of the first excited state transition, with a ten-time reduction
when passing from zero to half flux [see Figure 11(b,c)]. Indeed higher level
transitions have a strong flux dependence, consistent with the ground state
and first state delocalization over several Josephson potential wells.
Notably, this delocalization magnification is also linked to an enhanced
amplitude of zero-point-fluctuations of the phase across the KITE, which is
estimated as

$\varphi_{\mathrm{zpf},\mu}=\left(\frac{2E_{C}}{E_{L}}\mu^{2}\right)^{1/4}\;.$
(75)

From the experiment a factor $\varphi_{{\rm zpf},2}/\varphi_{{\rm zpf},1}\sim
2.1$ enhancement has been reported. The fact that this value is larger than
the expected $\sqrt{2}$ one, has been ascribed to the presence of double phase
slip events when the external flux is zero (in the conventional $2\pi$
periodicity regime) [173].

These results can have important fundamental consequences in finding systems
where charge and flux fluctuations are no more asymmetric. Indeed, the
expression for zero-point-fluctuations can be rewritten as

$\varphi_{\mathrm{zpf},\mu}=\sqrt{8\pi\alpha_{{\rm
ee}}\mu\frac{Z}{Z_{\mathrm{vac}}}}$ (76)

where $\alpha_{{\rm ee}}=1/137$ is the fine structure constant and $Z$,
$Z_{{\rm vac}}$ represent the circuit and vacuum impedance, respectively. We
indeed recall that the ratio between flux and charge fluctuations is of order
$8\alpha_{{\rm ee}}$.In addition, in a circuit where electrons can only move
in packets of $\mu$ Cooper pairs, one would expect that the ratio of flux to
charge fluctuations gains a factor of $\mu$. This, in turn, implies that more
complex superconducting circuits where transport is mediated solely by
correlated entities of multiplies of Cooper pairs will further lower the ratio
between flux and charge fluctuations, paving the way for a novel class of
superconducting devices with built-in symmetric noise response.

##  5 Conclusions

Thanks to their versatility and high-degree of control, superconducting
circuits are at the forefront of novel quantum technology applications. Among
all superconducting based qubits are a promising and powerful resource for the
ensuing quantum computing era. After the recent demonstration of working
devices able to perform non-trivial quantum computational tasks [27, 26], in
the so-called noise intermediate-scale quantum framework, worldwide
theoretical and experimental efforts are devoted in finding strategies to
completely suppress, or at least to mitigate, errors and noise sources. Here
we focused on novel circuit designs, i.e. multi-mode superconducting circuits,
that exploit their topology to achieve noise protection at a hardware level.
In essence, the larger number of degrees of freedom, with respect to
conventional single-mode qubits, is at the root of robust encoding of quantum
information against environmental disturbances. Different proof-of-concept
experiments have recently demonstrated the huge potential of such multi-mode
qubits, achieving performance comparable, or even greater than, state-of-the-
art single-mode superconducting qubits. One of the next key steps in building
quantum information platforms consist in translating noise protection on a
larger scale, together with the development of modular and versatile hardware
and circuit couplers that can guarantee efficient information transfer and
high-fidelity of different computational tasks [174, 175, 176]. A modular
approach can indeed manage complex systems by assembling smaller components
into larger networks, possibly performing operations in noisy resilient sub-
parts and transferring information via efficient communication channels.

An interesting perspective in this direction would be the integration of
superconducting circuit technology with hybrid superconductor/semiconductor
based devices [176, 177, 178]. Here, the large versatility offered by
semiconductors, both in geometrical control and in the possible gate
tunability, can indeed further enhance the performance of superconducting
based qubits in a already mature technological context.

## Acknowledgments

A.C. was supported by the Würzburg-Dresden Cluster of Excellence ct.qmat,
EXC2147, project-id 390858490, and the DFG (SFB 1170). A.C. also thanks the
Bavarian Ministry of Economic Affairs, Regional Development and Energy for
financial support within the High-Tech Agenda Project “Bausteine für das
Quanten Computing auf Basis topologischer Materialen”.

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# Procesi’s Conjecture on the Formanek-Weingarten Function is False

Maciej Dołęga  Institute of Mathematics, Polish Academy of Sciences, ul.
Śniadeckich 8, 00-956 Warszawa, Poland<EMAIL_ADDRESS>and  Jonathan Novak
Department of Mathematics, University of California, San Diego, USA
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we disprove a recent monotonicity conjecture of C. Procesi on
the generating function for monotone walks on the symmetric group, an object
which is equivalent to the Weingarten function of the unitary group.

MD is supported from Narodowe Centrum Nauki, grant 2021/42/E/ST1/00162/

JN is supported by NSF grant DMS 1812288 and a Lattimer Fellowship.

##  1\. Introduction

Let $\Gamma_{d}$ be the Cayley graph of the symmetric group $\mathrm{S}(d)$ as
generated by the conjugacy class of transpositions. Thus $\Gamma_{d}$ is a
${d\choose 2}$-regular graded graph with levels $L_{0},\dots,L_{d-1},$ where
$L_{k}$ is the set of permutations which factor into a product of $d-k$
disjoint cycles. Let us mark each edge of $\Gamma_{d}$ corresponding to the
transposition $(i\ j)$ with $j\in\\{2,\dots,d\\}$, the larger of the two
symbols interchanged. This edge labeling was first considered by Stanley [8]
and Biane [1] in connection with noncrossing partitions and parking functions.

A walk on $\Gamma_{d}$ is said to be _monotone_ if the labels of the edges it
traverses form a weakly increasing sequence. The combinatorics of such walks
has been intensively studied in recent years, beginning with the discovery [4]
that these trajectories play the role of Feynman diagrams for integration with
respect to Haar measure on unitary groups. This is part of a broader subject
nowadays known as Weingarten calculus, see [2].

Although non-obvious, it is a fact that the number of monotone walks of given
length $r$ between two given permutations $\rho,\sigma\in\mathrm{S}(d)$
depends only on the cycle type $\alpha\vdash d$ of the permutation
$\rho^{-1}\sigma$. It is therefore sufficient to consider the number
$m^{r}(\alpha)$ of $r$-step monotone walks on $\Gamma_{d}$ beginning at the
identity permutation and ending at a fixed permutation of cycle type $\alpha$.
To each partition $\alpha\vdash d$ we associate the generating function

(1.1) $M_{\alpha}(x)=\sum_{r=0}^{\infty}m^{r}(\alpha)x^{r}$

enumerating monotone walks on $\Gamma_{d}$ of arbitrary length and type
$\alpha$. It is known [3] that

(1.2) $M_{\alpha}(x)=\sum_{\lambda\vdash
d}\frac{\chi^{\lambda}_{\alpha}}{\prod_{\Box\in\lambda}h(\Box)(1-c(\Box)x)},$

where $\chi^{\lambda}_{\alpha}$ are the irreducible characters of the
symmetric group $\mathrm{S}(d)$, with $h(\Box)$ and $c(\Box)$ being,
respectively, the hook length and content of a given cell $\Box$ in the Young
diagram of $\lambda$ (see [7] for definitions). In particular, $M_{\alpha}(x)$
is a rational function of $x$ which may be considered as a continuous function
of $x$ on the interval $(0,\frac{1}{d-1})$ whose outputs are positive rational
numbers. Up to a simple rescaling, the values $M_{\alpha}(\frac{1}{N})$
coincide with the values of the Weingarten function of the unitary group
$\mathrm{U}(N)$; see [3, 4].

In a recent paper [5], Procesi has pointed out that the function
$M_{\alpha}(x)$ was also studied from the perspective of classical invariant
theory by Formanek, and that in this context the values
$M_{\alpha}(\frac{1}{d})$ have special significance. Procesi tabulated these
numbers for all diagrams $\alpha\vdash d\leq 8$, and on the basis of these
computations made the following conjecture.

######  Conjecture 1.1.

If $\alpha>\beta$ in lexicographic order, then
$M_{\alpha}(\frac{1}{d})>M_{\beta}(\frac{1}{d}).$

In this brief note we give explicit numerical examples which show that
Conjecture 1.1 is false.

##  2\. Small $x$

We first clarify that Procesi’s Conjecture 1.1 refers to lexicographic order
on partitions viewed as nondecreasing sequences of positive integers, with $1$
the first letter in the alphabet, $2$ the second letter, and so on. For
example, the partitions of six listed in lexicographic order are

$\displaystyle 111111$ $\displaystyle 11112$ $\displaystyle 1113$
$\displaystyle 1122$ $\displaystyle 114$ $\displaystyle 123$ $\displaystyle
15$ $\displaystyle 222$ $\displaystyle 24$ $\displaystyle 33$ $\displaystyle
6,$

and Conjecture 1.1 says that the numbers $M_{\alpha}(\frac{1}{6})$ strictly
decrease as $\alpha$ moves down this list, and this is so. However, the
pattern fails for sufficiently large degree $d$.

The first sign that Conjecture 1.1 might be false in general is that it is
incompatible with the known $x\to 0$ asymptotics of $M_{\alpha}(x)$. The
minimal length of a walk on $\Gamma_{d}$ from the identity to a permutation of
type $\alpha$ is $d-\ell(\alpha)$, and thus by parity the number
$m^{r}(\alpha)$ can only be positive when $r=d-\ell(\alpha)+2k$ with $k$ a
nonnegative integer. We may therefore reparameterize the counts
$m^{r}(\alpha)$ as $m_{k}(\alpha):=m^{d-\ell(\alpha)+2k}(\alpha)$ for
$k\in\mathbb{N}_{0}$. The generating function $M_{\alpha}(x)$ then becomes

(2.1)
$M_{\alpha}(x)=x^{d-\ell(\alpha)}\sum_{k=0}^{\infty}m_{k}(\alpha)x^{2k}.$

It is then clear that

(2.2) $\lim_{x\to 0}\frac{M_{\beta}(x)}{M_{\alpha}(x)}=0$

whenever $\ell(\alpha)>\ell(\beta)$, which is incompatible with lexicographic
order.

One might nevertheless hope that when we compare the small $x$ behavior of
$M_{\alpha}(x)$ and $M_{\beta}(x)$ with $\alpha$ and $\beta$ being partitions
of the same length, we find compatibility with lexicographic order. This too
is false, as can be seen from the fact [3] that

(2.3)
$m_{0}(\alpha)=\prod_{i=1}^{\ell(\alpha)}\operatorname{Cat}_{\alpha_{i}-1},$

where $\operatorname{Cat}_{n}=\frac{1}{n+1}{2n\choose n}$ is the Catalan
number. Then for $\alpha,\beta\vdash d$ partitions of the same length $\ell$,
we have

(2.4) $\lim_{x\to
0}\frac{M_{\beta}(x)}{M_{\alpha}(x)}=\prod_{i=1}^{\ell}\frac{\operatorname{Cat}_{\beta_{i}-1}}{\operatorname{Cat}_{\alpha_{i}-1}}.$

For small values of $d$, it does indeed appear to be the case that this
product is smaller than $1$ when $\alpha>\beta$, but this is a law of small
numbers. Consider the case where

(2.5) $\alpha=(1,\underbrace{3,\dots,3}_{n})\quad\text{ and
}\quad\beta=(\underbrace{2,\dots,2}_{n},n+1)$

Then $\alpha$ and $\beta$ are partitions of the same degree $d=3n+1$, they
have the same length $\ell(\alpha)=\ell(\beta)=n+1$, and $\alpha$ precedes
$\beta$ in the lexicographic order. However, the ratio of the corresponding
Catalan products tends to infinity as $n\to\infty$,

(2.6)
$\frac{\operatorname{Cat}_{n}}{2^{n}}\sim\frac{1}{\sqrt{\pi}n^{3/2}}\cdot
2^{n}.$

##  3\. Counterexamples

To give a counterexample to Conjecture 1.1 itself, we return to the character
formula (1.2), which in fact yields counterexamples if one goes a bit farther
than the data tabulated in [5]. Let $\alpha^{+}$ denote the successor of
$\alpha$ in the lexicographic order. The first value of $d$ for which
Conjecture 1.1 fails is the famously unlucky number $d=13$, for which there
exists precisely one violating pair $\alpha,\alpha^{+}$. This pair is

$M_{(1^{6},7)}\left(\frac{1}{13}\right)=\frac{13^{13}}{(13!)^{2}}\frac{30132115571}{1149266300}<\frac{13^{13}}{(13!)^{2}}\frac{426729597219}{16089728200}=M_{(1^{5},2^{4})}\left(\frac{1}{13}\right)$

We have tested Conjecture 1.1 for $d\leq 20$ and it fails for all $13\leq
d\leq 20$. Moreover the size of the set

$G_{d}:=\left\\{\alpha\vdash d\colon
M_{\alpha}\left(\frac{1}{d}\right)<M_{\alpha^{+}}\left(\frac{1}{d}\right)\right\\}$

of consecutive failures at rank $d$ increases with $d$. For instance

$\displaystyle G_{14}=\\{$ $\displaystyle(1^{7},7),(1^{5},2,7),(1^{5},9)\\},$
$\displaystyle G_{15}=\\{$
$\displaystyle(1^{8},7),(1^{6},2,7),(1^{6},9),(1^{4},11),(1^{3},2,10),(1^{3},3,9)\\},$
$\displaystyle G_{16}=\\{$
$\displaystyle(1^{11},5),(1^{9},7),(1^{7},2,7),(1^{7},9),(1^{6},10),(1^{5},2^{2},7),(1^{5},11),(1^{4},2,10),$
$\displaystyle(1^{4},3,9),(1^{3},13),(1,4,11)\\}.$

Even though Conjecture 1.1 seems to fail for all $d\geq 13$ the structure of
the failure set $G_{d}$ appears to be very interesting: it seems that when $d$
is large, the points in $G_{d}$ form many short lexicographic intervals and
one large lexicographic interval. For instance $|G_{20}|=45$, so the
proportion of the length of a typical interval on which
$M_{\alpha}(\frac{1}{|\alpha|})$ is monotone is equal to $\frac{1}{45}$.
Nevertheless, for the interval $((1,2^{2},4,11),(2,5,13)]$, whose cardinality
is equal to $151$, one has $((1,2^{2},4,11),(2,5,13)]\cap
G_{20}=\\{(2,5,13)\\}$. The number of partitions of size $20$ is $627$,
therefore there exists an interval on which $M_{\alpha}(\frac{1}{|\alpha|})$
is monotone and which is more than ten times longer than its expected length.
This suggests that a weaker version of Conjecture 1.1 might be true. Let
$\mathcal{P}_{d}$ denote the set of partitions of size $d$.

######  Question 3.1.

Is it true that there exists constant $C>0$ such that for every positive
integer $d$ there exists partitions $\alpha^{d}>\beta^{d}\in\mathcal{P}_{d}$
such that for every lexicographic sequence
$\alpha^{d}\geq\alpha>\beta\geq\beta^{d}$ we have
$M_{\alpha}(\frac{1}{d})>M_{\beta}(\frac{1}{d})$ and
$\frac{|[\alpha_{d},\beta_{d}]|}{|\mathcal{P}_{d}|}\geq C$?

We do not know the answer to this question and we leave it wide open. It would
also be very interesting to find an explicit description of the set $G_{d}$,
which appears to consists of very specific partitions which might be
classifiable. Even though Conjecture 1.1 turned out to be false we believe
that the research initiated by Procesi [5] on the behaviour of the function
$M_{\alpha}(\frac{1}{|\alpha|})$ merits further investigation. Indeed,
Procesi’s work has added a new and largely unexplored dimension to Weingarten
calculus.

## Acknowledgments

The SageMath computer algebra system [6] has been used for experimentation
leading up to the results presented in the paper.

## References

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