# Optimizing Nuclear Reaction Analysis (NRA) using Bayesian Experimental
Design

U. von Toussaint, T. Schwarz-Selinger, and S. Gori  Max-Planck-Institut für
Plasmaphysik, EURATOM Association, Boltzmannstr. 2, 85748 Garching, Germany

###### Abstract

Nuclear Reaction Analysis with 3He holds the promise to measure Deuterium
depth profiles up to large depths. However, the extraction of the depth
profile from the measured data is an ill-posed inversion problem. Here we
demonstrate how Bayesian Experimental Design can be used to optimize the
number of measurements as well as the measurement energies to maximize the
information gain. Comparison of the inversion properties of the optimized
design with standard settings reveals huge possible gains. Application of the
posterior sampling method allows to optimize the experimental settings
interactively during the measurement process.

###### pacs:

02.50.Le, 02.50.Tt, 25..55.-e, 29.85.Fj

##  I Introduction

The rising price for oil has recently shifted the focus to other possible
sources of energy, preferably without adverse effects to the environment. One
of the methods presently being developed is nuclear magnetic fusion. The
objective of fusion research is to harness the energy provided by the fusion
of hydrogen isotopes. In the fusion experiment ITER, presently under
construction in Cadarache, France, the necessary data to design and operate
anelectricity-producing plant shall be gained. ITER is a tokamak, an
intermittent operating device in which strong magnetic fields confine a torus-
shaped plasma.

Figure 1: In-vessel view of ITER. The surface of the main chamber is
Beryllium, the material of the divertor (in lower part of the vacuum vessel)
is carbon and tungsten. Source:Iter08 , published with permission of ITER.

Since the confinement is not perfect (and must not be) there are always
interactions between the plasma and the plasma-facing (wall) components (PFCs)
which have to be taken into account. One of the key aspects in the licensing
process of ITER is a strict upper limit of the total amount of radioactive
tritium accumulated in the vessel walls, which is presently at 700g
tritiumIter08 . The prediction of the amount of retained tritium is
complicated by the material choice of ITER (Fig.1): The main vessel walls are
Beryllium, the strike-points are made of carbon (CFC) and the other parts of
the divertor are tungsten. During the operation of ITER the interaction of the
plasma and high energy 14MeV-neutrons with the vessel walls will lead to
erosion, redeposition, material mixing and alloy formation. Since even the
hydrogen retention properties of pure materials are still subject to current
research, a significant amount of additional experimental data is required to
develop and calibrate the theoretical models which will be needed to process
the huge number of material combinations created in ITER.

However, even the first step - measuring hydrogen depth profiles in material
composites - is challenging for many reasons; here we will mention only two:
a) Hydrogen and its isotopes are very volatile, which can easily distort
measurements of depth profiles and b) Hydrogen is usually the main component
of the residual gas in vacuum chambers which precludes the use of many well-
established analysis methods.

One method which holds great promise to overcome these difficulties is the
Nuclear Reaction Analysis (NRA) of deuterium using 3He as probing particle. It
is a specific and sensitive method, and has a sufficient analysis depth.
However every data point takes about 30min to measure and the extraction of
the concentration depth profile is an ill-posedinversion problem requiring the
deconvolution of the measured data vector, here even more challenging than in
Rutherford BackscatteringToussaint2000 . Therefore the experimental setup (ie
the choice of the analysis energies) should provide a maximum of information.
So far the most common choice of the 3He energies for the measurements was
simply equidistant. Using Bayesian Experimental Design the performance of the
method can be improved considerably (in some cases up to orders of magnitude)
and quantitative measures can bederived about the expected utility of further
(time consuming) measurements. sectionNuclear Reaction Analysis The basic
principle of Nuclear Reaction Analysis is straightforward: The sample is
subjected to an energetic ion beam (here 3He) with initial energy
${E}_{i}^{0}$ and incoming angle $\phi$, which reacts predominately with the
species of interest (Deuterium) and the products of the reaction are measured
under a specified angle $\theta$. Given the total number of impinging ions Ni,
the energy dependent cross-section of the reaction $\sigma\left(E\right)$, the
efficency of the detection and the geometry of the set-up $\mu$ the measured
total signal counts $d_{i}$ depend (in the limit of small concentrations)
linearly on the concentration profile $c\left(x\right)$ of the species in the
depth $x$:

$d_{i}=d\left(E_{i}^{0}\right)=\mu\mathrm{N}_{i}\int_{0}^{x\left(E_{i}^{0}\right)}\\!dx\,\sigma\left(E\right)c\left(x\right)=\mu\mathrm{N}_{i}\int_{0}^{x\left(E_{i}^{0}\right)}\\!dx\,\sigma\left(E\left(x,E_{i}^{0}\right)\right)c\left(x\right)+\epsilon_{i},$
(1)

where $\epsilon_{i}\sim{\phantom{a}}N\left(0,\sigma_{i}\right)$ represents
normal distributed noise. Repeated measurements with different initial
energies of 3He provide increasing information about the Deuterium depth
profile. The question addressed in the following is: Given a set of already
measured data $d\left(E_{i}^{0}\right)$ which measurement energy should be
chosen next?

To evaluate Eq. 1 we first need to specify the cross section
$\sigma\left(E\right)$ and the energy $E\left(x\right)$ of the incident
particle on its path through the sample.

###  I.1 Cross-Section

The relevant cross-section for the reaction D+${}^{3}\mathrm{He}\rightarrow
p+{}^{4}\mathrm{He}$\+ 18.352 MeV (in standard notation written as
$\mathrm{D}\left({}^{3}\mathrm{He},p\right){}^{4}\mathrm{He}$) has been
(re-)measured recently Alimov05  in the range of 550 keV to 6MeV and the
obtained cross-section values have been given in tabular form. Using the same
method as Alimov05  we added several cross-section measurements at energies
below 690keV and fitted both data sets taking into account also earlier
measurements Moeller80 ; Bosch92

$\sigma\left(E\left[MeV\right]\right)=829.98*\frac{E^{2.83962}\left(0.270713*e^{-2.2158E}+0.0182765\right)}{E^{3.47626}+0.270713*e^{-1.17229E}-0.00123669}\left[mb\right].$
(2)

using the Levenberg-Marquardt algorithm minimizing the $\chi^{2}$-misfit with
the variance set to $d_{i}$.

Figure 2: a) Differential cross-section of the nuclear reaction D(3He,p)4He in
the laboratory system with a reaction energy of Q=18.352 MeV (left). b) On the
right hand side the typical results for an NRA measurement are shown
(simulated data of a tungsten sample with an exponentially decaying D
concentration profile (cf. Eq.4). The uncertainties due to the counting
statistics are usually dominated by the uncertainty of the analysis current.

The cross-section is plotted in Fig.2a and shows a broad maximum around 630
keV and is above 3 MeV nearly constant at 8 mb/sr up to 6 MeV (above that
there are no data available). The reaction energy is very high (Q=18.352 MeV)
and most of the energy is transferred to the resulting proton. This leads to a
very good S/N-ratio of the measurement because other particles can easily be
separated by energy.

###  I.2 Energy Loss

The energy loss of the impinging 3He-ion in the sample is determined by the
stopping power $S(E)$ of the sample

$\frac{dE}{dx}=-S(E),$ (3)

which can be solved to get the depth dependent energy $E_{i}(x)$ for different
initial energies ${E}_{i}^{0}$. Parameterizations and tables of $S$ for
different elements are given in Tesmer95 . Since the amount of hydrogen in the
sample is usually well below $1\%$ (with the exception of a very thin surface
layer), the influence of the hydrogen concentration on the stopping power can
be neglected in most cases.

###  I.3 Simulation of Mock Data

To simulate mock data for typical accelerator parameters a tungsten sample
$\left(\rho=19.3\mathrm{g/cm}^{3}\right)$ with a (high) surface concentration
of 12$\%$ Deuterium, followed by an exponentially decaying Deuterium
concentration down to a constant background level, described by

$c(x)=a_{0}*\exp\left(-\frac{x}{a_{1}}\right)+a_{2}=0.1\exp\left(-\frac{x}{2.5*10^{18}\frac{\mathrm{at}}{\mathrm{cm}^{2}}}\right)+0.02$
(4)

has been used 111Generally ion-beam analysis methods are sensitive to the
areal density of the species $\left(\mathrm{at/cm^{2}}\right)$ which can be
converted into a depth scale if the density of the material is known. In
tungsten $a_{1}=2.5*10^{18}\frac{\mathrm{at}}{\mathrm{cm}^{2}}$ correspond to
$a_{1}=$395nm.. The corresponding mock data for a set of initial energies
E0={500, 700, 1000, 1300, 1600, 2000, 2500, 3000}keV is shown in Fig. 2b. The
variations in the detected yields display the interplay of the increasing
range of the ions with increasing energy and the reduced cross-section at
higher energies modulated with the decreasing Deuterium concentration at
larger depths. The increase of the signal by raising the initial energy from
2500 keV to 3000 keV is already caused by the constant Deuterium background of
2%. The accelerator time which would be needed to obtain the 8 data points is
around one working day taking into account the necessary interleaved
calibration measurements:The ion bombardment causes an energy and depth
dependent loss of Deuterium. Commonly a first order correction is applied by
normalizing the yields with respect to the yields obtained from repeated
calibration measurements using the same (typically low, e.g. 690keV) initial
energy222Remark: This approach of taking into account the Deuterium loss,
although in widespread use, will almost always introduce a systematic bias in
the derived concentration profile: The loss of Deuterium near the surface is
used to correct the signal resulting from the overall Deuterium concentration
profile, where the losses are usually different..

The uncertainty of the detector is given by Poisson-statistics. However,
fluctuations in the beam current measurements are very often the dominating
factor, affecting the pre-factor $\mathrm{N}_{i}$ in Eq.1. An accuracy of up
to 3$\%$ can be achieved (e.g. by using the number of Rutherford-scattered 3He
ions on a thin gold-coating on top of the sample as reference). The error of
the renormalization procedure is harder to quantify. For simplicity we will
use $\sigma_{i}=\mathrm{max}\left(5\%d_{i},\sqrt{d_{i}}\right)$ as uncertainty
of the data in the following, acknowledging that there is room forimprovement.

##  II Bayesian Experimental Design

Bayesian Experimental Design (BED) offers the tempting possibility to actively
select (and optimize) the experimental parameters for the next measurement(s)
based on objective criteria. Especially if measurements are expensive or time
consuming (like in the case of energy changes of an accelerator) it is a huge
advantage to know where to look next, so as to learn as much as possible. The
problem of experimental design has already been studied by Lindley back in
1956 Lindley56  in a Bayesian setting and Fedorov published his influential
book in 1972 Fedorov72  \- but the limitations in computational power limited
the application of experimental design almost always to simple (linear)
problems. This situation changed in the recent years and consequently there is
a renewed interest to apply BED also to (non-linear) real-world problems (see
e.g. Loredo03  and references therein or e.g. Preuss08 ; Dreier07 ; Fischer04
; Caticha00 . Not surprising the interest is biggest in branches of physics
where the experimental possibilities are severely restricted: Astronomy,
Fusion research,… Given the excellent account of BED in Loredo03  we only
summarize the key principles: In a first step an appropriate utility function
U has to be agreed upon: It describes the value which we assign to the
measurement results of an experiment and may include parameters like costs of
an experiment, duration, parameter uncertainty etc. Several utility functions
are considered in Chaloner95 . With focus on parameter estimation it was
proposed Lindley56 ; Bernardo79  to use the Kullback-Leibler divergence (KLd)
between the posterior and the prior distributions as utility function. The KLd
for a new datum D is given by

$U_{KL}\left(D,\underline{d},\eta\right)=\int\\!d\underline{\alpha}\;p\left(\underline{\alpha}|D,\underline{d},\eta\right)\log\left[\frac{p\left(\underline{\alpha}|D,\underline{d},\eta\right)}{p\left(\underline{\alpha}|\underline{d},\eta\right)}\right].$
(5)

Next we try to identify the action $\eta$ which maximizes the expected
utility. ’Expected’ utility because we have to account for the prediction
uncertainty for $D$. To compute the expected utility (EU) we have to average
over the new datum D weighted by the marginal likelihood for the new datum
given the observation of the old data $\underline{d}$

$\displaystyle EU\left(\underline{d},\eta\right)$ $\displaystyle=$
$\displaystyle\int\\!dD\;p\left(D|\underline{d},\eta\right)\cdot
U_{KL}\left(D,\underline{d},\eta\right)$ (6) $\displaystyle=$
$\displaystyle\int\\!dD\;p\left(D|\underline{d},\eta\right)\int\\!d\underline{\alpha}\;p\left(\underline{\alpha}|D,\underline{d},\eta\right)\log\left[\frac{p\left(\underline{\alpha}|D,\underline{d},\eta\right)}{p\left(\underline{\alpha}|\underline{d},\eta\right)}\right]$
$\displaystyle=$
$\displaystyle\int\\!dD\;p\left(D|\underline{d},\eta\right)\int\\!d\underline{\alpha}\;\frac{p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d},\eta\right)}{p\left(D|\underline{d},\eta\right)}\log\left[\frac{p\left(D|\alpha,\underline{d},\eta\right)p\left(\alpha|\underline{d},\eta\right)}{p\left(\underline{\alpha}|\underline{d},\eta\right)p\left(D|\underline{d},\eta\right)}\right]$
$\displaystyle=$
$\displaystyle\int\\!dD\;\int\\!d\underline{\alpha}\;p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d},\eta\right)\log\left[\frac{p\left(D|\underline{\alpha},\underline{d},\eta\right)}{p\left(D|\underline{d},\eta\right)}\right]$
$\displaystyle=$
$\displaystyle\int\\!dD\;\int\\!d\underline{\alpha}\;p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d},\right)\log\left[\frac{p\left(D|\underline{\alpha},\underline{d},\eta\right)}{\int\\!d\underline{\alpha}\;p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d}\right)}\right]$

where we dropped the $\eta-$dependence of the posterior of
$\underline{\alpha}$ in the last line, since our knowledge about
$\underline{\alpha}$ is not influenced by a possible future action. Closer
inspection of Eq. 6 reveals that only two different probability distributions
are required to compute the expected utility: the posterior distribution of
$\underline{\alpha}$ given the old data $\underline{d}$,
$p\left(\underline{\alpha}|\underline{d}\right)$ and the likelihood of the new
datum $D$ based on the previous measurements,
$p\left(D|\underline{\alpha},\underline{d},\eta\right)$.

###  II.1 The Linear Design

Assuming that the concentration profile $c(x)$ depends linearly on the
concentrations $c_{i}\left(x_{i}\right),i=1..q$ at a given set of $q$ support
points $\underline{x}$ then Eq.1 can be recast in the following form

$\underline{d}=\underline{f}+\underline{\epsilon}=\underline{\underline{M}}\,\underline{c}+\underline{\epsilon},$
(7)

where the data vector $\underline{d}$ is of size $p$, the matrix
$\underline{\underline{M}}$ is a $p\mathrm{x}q-$matrix and the parameter-
vector $\underline{c}$ has $q$ components. However, the requirement of
linearity applies only to the concentration parameter vector
${\underline{c}}$, the functional form of the concentration may be much more
complex, e.g. $c(x)=c_{1}*\left(x-x_{3}\right)^{4}+c_{2}*\sqrt{|x-x_{1}|}$,
although almost always $c(x)$ is chosen to be constant between the different
support points: $c(x)=c_{i},\forall x\in\left[x_{i},x_{i+1}\right]$ or as
linear interpolation between the support points. The noise vector
$\underline{\epsilon}$ is normally distributed
$\underline{\epsilon}\sim{\phantom{a}}N\left(0,\underline{\underline{\Sigma^{-1}}}\right)$,
where $\underline{\underline{\Sigma}}$ is a diagonal matrix with the entries
$\underline{\underline{\Sigma}}_{ii}=1/\sigma_{i}^{2}$. Every row of
$\underline{\underline{M}}_{j}$ is given by the solution of Eq. 1 for a
specified initial energy $E_{j}^{0}$,
${\underline{m}\left(E_{j}^{0}\right)}^{T}$. The consideration of the
uncertainties in the entries of the matrix due to energy straggling of the
impinging particles is beyond the scope of the present paper, but see e.g.
Mayer08 .

With a Gaussian likelihood for the existing data and a flat prior for the
parameters the posterior of the concentration vector reads

$\displaystyle p\left(\underline{c}|\underline{d},\eta\right)\propto
p\left(\underline{d}|\underline{c},\eta\right)$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\exp\left(-\frac{1}{2}\left(\underline{d}-\underline{\underline{M}}\,\underline{c}\right)^{T}\underline{\underline{\Sigma}}\left(\underline{d}-\underline{\underline{M}}\,\underline{c}\right)\right)$
(8) $\displaystyle=$
$\displaystyle\sqrt{\frac{\det{\underline{\underline{A}}}}{\left(2\pi\right)^{q}}}\exp\left(-\frac{1}{2}\left(\underline{c}-\underline{c_{0}}\right)^{T}\underline{\underline{A}}\left(\underline{c}-\underline{c_{0}}\right)\right)$

with

$\underline{\underline{A}}={\underline{\underline{M}}}^{T}\underline{\underline{\Sigma}}\,\underline{\underline{M}}\;\;\;\;\mathrm{and}\;\;\;\;\underline{c_{0}}={\underline{\underline{A}}}^{-1}{\underline{\underline{M}}}^{T}\underline{\underline{\Sigma}}\,\underline{d}.$
(9)

The posterior distribution of $\underline{c}$ including the new data point $D$
with its uncertainty $\sigma$,
$p\left(\underline{c}|D,\underline{d},\eta\right)$ can similarly be cast in a
Gaussian form. Therefore Eq. 6 can be solved analytically Fedorov72  and
yields a simple closed form for the exponential utility Dreier07 ; Preuss08 :

$\mathrm{EU}\left(\underline{d},\eta\right)=\frac{1}{2}\left(\log\left(1+\mathrm{G}\right)-r\right)$
(10)

with

$G=\frac{\underline{m}\left(\eta\right)^{T}\underline{\underline{A}}^{-1}\underline{m}\left(\eta\right)}{\sigma^{2}}.$
(11)

If $p\left(D|\underline{c},\eta\right)$ is Gaussian then $r=0$. The variation
of the EU depends on the vector $\underline{m}\left(\eta\right)$ which in turn
is uniquely determined by the choice of the initial energy $E_{p+1}^{0}$. The
optimum (maximum of the EU) can be found by a simple 1-D scan of the energy.

The sequential design approach in action is displayed in Fig 3. Starting from
the surface the concentration at increasingly larger depth intervals is of
interest. For this example the chosen depths are 0 nm, 80 nm, 240 nm, 470 nm
and 950 nm. After initial measurements at 400 keV, 700 keV and 3000 keV
(representing the lower and upper limit of the useful energy range for the
measurements and one calibration measurement) the best energy for the next
measurement has to be determined. The EU for this first cycle has a maximum at
1250keV (solid line). After a measurement with this energy the EU for the next
measurement has its maximum at 960keV and about twice the EU than before.
This, on the first glance, surprising increase of the EU can be made
transparent: With 5 unknowns and 5 (informative) measurements the solution
space of this linear problem no longer covers a sub-manifold of the parameter
space: It ’collapses’ and the volume of the ’occupied’ parameter space starts
to be determined by the measurement uncertainties. Therefore the 5-th
measurement has a very high EU. In the following cycle(s) the amplitude of the
EU is much lower since the subsequent measurements now gradually shrink the
’volume’ of the parameter posterior distribution. As long as the EU is above
the intended threshold for new measurements (which depends on the addressed
physical problem) further measurements are indicated.

How much better is the BED derived experiment compared to an experiment with
the same number but equidistant chosen initial energies? The entropy of the
parameter posterior distribution would be the obvious quantity to compare.
However, for the time being, many scientists are not happy with this measure
and prefer a more familiar measure, e.g. the condition number. The condition
number of the (pseudo-)inverse of $\underline{\underline{M}}$ is often used to
characterize linear least squares problems numrecipes  and is a measure how
strongly uncertainties in the data vector $\underline{d}$ may be amplified by
multiplication with the pseudo-inverse matrix. Using this measure the BED
optimized setting surpasses the equidistant experiment by a factor of more
than 100 (!).

Figure 3: Expected Utility for subsequent measurements. All measurements are
performed with the initial energy $E^{0}$ suggested by the maximum of the EU
in the corresponding cycle (indicated by marks on the energy axis).

###  II.2 Non-linear Design

The analytical solution in the preceeding case was possible because several
approximations have been applied: The integration range of the integration
over the predicted datum (a positive quantity) had to be changed from
$\int_{0}^{\infty}\\!dD$ to $\int_{-\infty}^{\infty}\\!dD$. Given the actual
number of counts and the uncertainties this can easily be justified.
Unfortunately, a similar change of the integration limits had to be applied
also in the parameter integration (from $\int_{0}^{1}\\!d\underline{c}$ to
$\int_{-\infty}^{+\infty}\\!d\underline{c}$) and here it definitely affects
the results. The analysis could be repeated substituting the analytical
integration by the numerical counterparts (e.g. using codes like VEGAS
numrecipes  or MCMC approaches). Furthermore, the uncertainty of the predicted
datum D is not constant but proportional to the signal $\sigma_{D}\propto D$
and therefore also the integrations over the data space have to be done
numerically. Under those circumstances there is no difference in the
computation to a non-linear experimental design problem. Additionally it
turned out that the actual quantity of interest is the decay length of the
hydrogen depth profile and that quite accurate data for the surface hydrogen
concentration are available (additionally measuring the ${}^{4}\mathrm{He}$ of
the $\mathrm{D}\left({}^{3}\mathrm{He},p\right){}^{4}\mathrm{He}$ reaction).
Therefore the optimal energy settings for the estimation of the parameters
$a_{1}$ and $a_{2}$ of concentration profiles of the functional form of Eq. 4
have to be computed. However, in non-linear experimental design the measured
data influences the EU (in contrast to the linear case: the maximum of the EU
is independent of the actually measured data, cf. Eq. 10) and this poses a
practical problem: The next accelerator energy has to be determined after the
previous measurement. And longer computation times to optimize the EU, causing
delays, are not tolerable.

Here the posterior sampling approach, suggested in Loredo03 , proved very
valuable. It turned out that sets of posterior samples
$p\left(a_{1i},a_{2i}|\underline{d},\eta\right),i=1..N$ drawn from
$p\left(a_{1},a_{2}|\underline{d},\eta\right)$ could be generated quite
efficiently (partly due to the low dimensionality of the parameter space).
With that sample (typically of size 1000) the denominator of the logarithm in
Eq. 6 is given by a simple summation

$\int\\!d\underline{\alpha}\,p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d}\right)\approx\sum_{i=1}^{N}p\left(D|\underline{\alpha_{i}},\underline{d},\eta\right).$
(12)

The biggest saving comes from the fact that the posterior sample is
independent from the actual value of $D$ and of the design action $\eta$: all
computations are reduced to repeated evaluations of the likelihood, which can
efficiently be vectorized. Finding the best energy is a matter of less than 5
minutes(!) on contemporary hardware (Linux-PC, 2GHz).

In Fig. 4 three cycles of the non-linear BED are shown: After a first
measurement at 500keV the posterior distribution of
$\left\\{a_{1},a_{2}\right\\}$ is visualized in the upper left graph by the
posterior sample. The single measurement does not allow to distinguish between
a large decay constant $a_{1}$ and low constant offset $a_{2}$ or vice versa.
The EU, plotted in the upper right graph, favors now a measurement at the
other end of the energy range (the maximum of the utility function is
encircled). After a measurement with 3MeV ${}^{3}\mathrm{He}$ the ’area’ of
the posterior distribution is significantly reduced (middle row, left graph):
The background concentration is below 3% but the decay length is still quite
undetermined. The EU has a maximum at 1500 keV, still with a pretty high EU.
Performing a measurement with 1500keV localizes the posterior distribution
around the true (but unknown value of $a_{1}=395$nm and $a_{2}=0.02$). The
next measurement should be performed at 1200keV but the EU is significantly
lower than before: subsequent measurements are predominantly improving the
statistics: a second measurement at 3 MeV provides nearly the same
information.

Figure 4: Three cycles of the Experimental Design process: On the left hand
side 1000 samples drawn from the posterior distribution
$p\left(a_{1},a_{2}|\underline{d},\eta\right)$ are displayed. On the right
hand side the EU is plotted and the maximum is indicated by a circle. The
corresponding abscissa value is the suggested next measurement energy.
Performingthat measurement yields the posterior distribution given in the next
row.

##  III Conclusion and Outlook

The concept of Bayesian Experimental Design allows to objectively optimize
experimental designs. Here we presented two different approaches to optimize
NRA depth profiling: First in a linear setting, allowing an analytical
solution and straightforward parametric studies. Second, a time-critical non-
linear experimental design problem which could be tackled using posterior
sampling. Both optimization procedures may considerably increase the accuracy
of the derived depth profiles compared to the present approach and at the same
time reduce the overall measurement time by signaling a diminishing utility of
further measurements. With the posterior sampling approach many sequential
measurements can now be optimized on the fly: This opens up the door for a
wealth of new applications of BED in the field of ion beam
analysisToussaint2000  as well as in other physical areas Loredo03 ; Preuss08
; Toussaint2006

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# Fractal percolation on statistically self-affine carpets

Kenneth Falconer and Tianyi Feng
Mathematical Institute, University of St Andrews, North Haugh, St Andrews,
Fife, KY16 9SS, Scotland

###### Abstract

We consider a random self-affine carpet $F$ based on an $n\times m$
subdivision of rectangles and a probability $0<p<1$. Starting by dividing
$[0,1]^{2}$ into an $n\times m$ grid of rectangles and selecting these
independently with probability $p$, we then divide the selected rectangles
into $n\times m$ subrectangles which are again selected with probability $p$;
we continue in this way to obtain a statistically self-affine set $F$. We are
particularly interested in topological properties of $F$. We show that the
critical value of $p$ above which there is a positive probability that $F$
connects the left and right edges of $[0,1]^{2}$ is the same as the critical
value for $F$ to connect the top and bottom edges of $[0,1]^{2}$. Once this is
established we derive various topological properties of $F$ analogous to those
known for self-similar carpets.

##  1 Introduction

The notion of fractal percolation was introduced in 1974 by Mandelbrot [12]
and discussed in his classic book [13], where it is termed ‘canonical
curdling’. Fractal percolation concerns the topological properties of a
statistically self-similar set, obtained by repeating a random operation at
ever-decreasing scales.

The square-based fractal percolation model, which has been well-studied [4, 5,
6, 14], is based on repeated subdivision of the unit square into smaller
squares which are selected at random. Let $m\geq 2$ be an integer and $0\leq
p\leq 1$ a probability. The unit square $E_{0}=[0,1]^{2}$ is divided into
$m^{2}$ closed subsquares of side $m^{-1}$ in the obvious way and each of
these subsquares is selected independently with probability $p$; we write
$E_{1}$ for the random set formed as the union of these selected squares. We
next divide each of these selected subsquares into squares of side $m^{-2}$
the same way, and again select these independently with probability $p$ to
form $E_{2}$. We continue in this way to obtain a decreasing sequence of
closed sets $E_{k}$ where each $E_{k}$ is a union of squares of side $m^{-k}$.
The random closed set $F:=\bigcap_{k=0}^{\infty}E_{k}$ is termed a
statistically self-similar carpet. For $p$ sufficiently large there is a
positive probability that $F$ is non-empty and, conditional on this, various
properties of $F$ have been examined, including the almost sure box-counting
and Hausdorff dimensions of $F$. Of particular interest here the topological
properties of $F$ which change dramatically as $p$ increases through a
critical value $p_{C}$. For $p<p_{C}$ the random set $F$ is almost surely
totally disconnected, whereas if $p\geq p_{C}$ it almost surely has many non-
trivial connected components and there is a positive probability of the left
and right sides of the square being joined through $F$, and similarly for the
top and bottom sides. Finding the value of $p_{C}$ even for small $m$ is
difficult, but some (not very close) lower and upper bounds have been
obtained, see [4, 6, 19].

Rather more general statistically self-similar sets have been considered,
particularly from a dimensional viewpoint, see for example [8, 15, 17]. This
paper is concerned with statistically self-affine carpets. These are
constructed, for integers $m>n\geq 2$ and a probability $0\leq p\leq 1$, in a
similar way to the self-similar square-based model above, except we select
subrectangles rather than subsquares. Thus $E_{0}=[0,1]^{2}$ is divided into
an $n\times m$ array of rectangles of sides $n^{-1}\times m^{-1}$ which are
selected independently with probability $p$, the selected rectangles being
repeatedly subdivided and selected in the same way, to get a decreasing
sequence of sets $E_{k}$ formed as a union of $n^{-k}\times m^{-k}$
rectangles, to yield a statistically self-affine set
$F=\bigcap_{k=0}^{\infty}E_{k}$. A more precise description of this model will
be given in the next section.

$\qquad E_{0}=[0,1]^{2}\qquad\qquad\quad\quad E_{1}\qquad\qquad\qquad\qquad
E_{2}\qquad\qquad\qquad\quad\ E_{3}\\!\qquad$

Figure 1: The first three steps in the construction of a statistically self-
affine $F$.

We are interested in topological properties of this statistically self-affine
$F$ and in particular to what extent a random construction based on hierarchy
of rectangles differs from the square case. With the rectangles becoming wide
and squat for large $k$, it is not clear that the critical probability for
horizontal (left-right) crossings to occur in $F$ is the same as that for
vertical (top-bottom) crossings. We will show that these critical
probabilities are infact equal, that is if for some $p$ there is a positive
probability of a horizontal crossing then there is also a positive probability
of a vertical crossing, and vice-versa. Once this is established, modifying
methods used for the statistically self-similar $m\times m$ case yields
information about the topology of the random set $F$.

We have aimed to make this article fairly self-contained by including some
proofs that are reminiscent of arguments that have been used for percolation
on statistically self-similar carpets.

##  2 Model and results

Here we describe the construction of the random set $F$ and state our results.
We fix integers $m>n\geq 2$ throughout the construction and its analysis.

We divide the unit square $E_{0}:=[0,1]^{2}$ into a rectangular array of $mn$
level-1 closed rectangles of sides $n^{-1}\times m^{-1}$ which we label (in
some order) $\\{R_{i_{1}}\\}_{1\leq i_{1}\leq mn}$. We then divide each
$\\{R_{i_{1}}\\}$ into $mn$ level-2 closed rectangles
$\\{R_{i_{1},i_{2}}\\}_{1\leq i_{2}\leq mn}$ of sides $n^{-2}\times m^{-2}$.
We continue in this way, so that $\\{R_{i_{1},\ldots,i_{k}}\\}_{1\leq
i_{1},\ldots,i_{k}\leq mn}$ are $(mn)^{k}$ level-$k$ closed rectangles of
sides $n^{-k}\times m^{-k}$, with $R_{i_{1},\ldots,i_{k}}=\bigcup_{1\leq i\leq
mn}R_{i_{1},\ldots,i_{k},i}$ for each $k$ and $(i_{1},\ldots,i_{k})$.

Let $0\leq p\leq 1$ be a probability. We select each rectangle
$\\{R_{i_{1}}\\}$ independently with probability $p$ and reject it with
probability $1-p$ and we write $E_{1}$ for the union of these selected
rectangles. For each of the selected rectangles $\\{R_{i_{1}}\\}$ we select
level-2 subrectangles $\\{R_{i_{1},i_{2}}\\}$ with probability $p$ and let
$E_{2}$ be the union of all these selected rectangles. We continue in this way
to get a decreasing sequence of closed sets $E_{k}$, the union of a random
selection of $n^{-k}\times m^{-k}$ closed rectangles, and let
$F=\bigcap_{k=1}^{\infty}E_{k}$ be the random limit set which will be closed.
The number of rectangles in $E_{k}$ is given by a Galton-Watson branching
process which implies in particular that there is a positive probability of
$F$ being non-empty if and only if $p>1/mn$.

We formalise this by defining a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ on the underlying rectangles by

$\Omega={\mathcal{P}}\Big{(}\big{(}\\{R_{i_{1}}\\}_{1\leq i_{1}\leq
mn},\\{R_{i_{1},i_{2}}\\}_{1\leq i_{1},i_{2}\leq mn},\ldots\big{)}\Big{)}$
(2.1)

where $\mathcal{P}$ denotes the power set, with $\mathcal{F}$ the product
$\sigma$-field on this countable sequence of sets, and the product probability
$\mathbb{P}$ defined by setting $\mathbb{P}(R_{i_{1},\ldots,i_{k}})=p$
independently for all $k$ and all $(i_{1},\ldots,i_{k})\ (1\leq i_{j}\leq
mn)$, and extending to $\mathcal{F}$. We identify
$R_{i_{1},\ldots,i_{k}}\in\omega$ with the event ‘the rectangle
$R_{i_{1},\ldots,i_{k}}$ is selected’ in the random geometric construction.

Then for $k\geq 1$,

$E_{k}=\bigcup\\{R_{i_{1},\ldots,i_{k}}:R_{i_{1}},R_{i_{1},i_{2}},\ldots.,R_{i_{1},\ldots,i_{k}}\in\omega\\}=\bigcap_{j=1}^{k}\bigcup\\{R_{i_{1},\ldots,i_{j}}\in\omega\\}\quad(\omega\in\Omega),$

and the random self-affine set $F$ obtained by this process is.

$F\equiv
F(\omega)=\bigcap_{k=1}^{\infty}E_{k}=\bigcap_{j=1}^{\infty}\bigcup\\{R_{i_{1},\ldots,i_{j}}\in\omega\\}\quad(\omega\in\Omega)$

Note that it is convenient to consider selection or rejection of all the
rectangles in the space, even if their ‘parent’ rectangle has not been
selected. We also let

$F_{k}=\bigcap_{j=k}^{\infty}\bigcup\\{R_{i_{1},\ldots,i_{j}}\in\omega\\}\quad(\omega\in\Omega);$

thus $F_{1}=F$ and, for $k\geq 1$, $F_{k}$ is the random set obtained starting
the random selection at level $k$, and assuming that all rectangle at levels
$1$ to $k-1$ are selected. Note that the sets $E_{k}$ and thus $F$ and $F_{k}$
are compact.

We will be interested in topological properties and especially connectivity
properties of $F$ and how they vary with $p$, see Figure 2. In particular we
consider when there is a positive probability of $F$ providing vertical and/or
horizontal crossings of the unit square, that is when there are points in
opposite sides of the square that are in the same connected component $F$.

(a) $p=0.4$ (b) $p=0.5$

(c) $p=0.78$

Figure 2: Realisations of $F$ based on a $3\times 4$ rectangular grid, for
different retention probabilities $p$.

We review first some basic properties of statistically self-affine sets. With
$F$ constructed as above, the following proposition records the probability of
$F$ being empty, and the almost sure box-counting, Hausdorff and Assouad
dimensions of $F$ if it is non-empty.

######  Proposition 2.1.

$({\rm i})$ If $0\leq p\leq 1/mn$ then $F=\emptyset$ almost surely. If
$1/mn<p\leq 1$ then $\mathbb{P}(F\neq\emptyset)=t>0$, where $t$ is the least
non-negative solution of $t=(pt+1-p)^{mn}$.

$({\rm ii})$ If $1/mn<p\leq 1$ then, conditional on $F\neq\emptyset$,

$\dim_{H}F=\dim_{B}F=\left\\{\begin{array}[]{ll}\log(pnm)\big{/}\log
n&(1/mn<p\leq 1/m)\\\ \log(pm^{2})\big{/}\log m&(1/m<p\leq
1)\end{array}\right.,$ $\dim_{A}F=2\ \ (1/mn<p\leq 1).$

The proof of Proposition 2.1 is sketched in Section 3.1.

Let $F$ be a subset of $\mathbb{R}^{2}$. We say that two points ${\bf
x}_{1},{\bf x}_{2}\in F$ are connected or joined, or that ${\bf x}_{1}$ is
connected to ${\bf x}_{2}$ in $F$ if they are in the same connected component
of $F$ (with respect to the usual Euclidean topology). We say that two sets
$A,B\subset\mathbb{R}^{2}$ are connected or joined in $F$ if there are points
${\bf x}_{1}\in A$ and ${\bf x}_{2}\in B$ that are connected in $F$.

The set $F$ is a horizontal or H-crossing of the unit square $[0,1]^{2}$ if
$\\{0\\}\times[0,1]$ is connected to $\\{1\\}\times[0,1]$. Similarly $F$ is a
vertical or V-crossing if $[0,1]\times\\{0\\}$ is connected to
$[0,1]\times\\{1\\}$. For a probability $0\leq p\leq 1$ we write
$\theta_{H}(p)$ and $\theta_{V}(p)$ for the probabilities that $F$ has a
horizontal or vertical crossing respectively. The following proposition, which
follows by a similar argument to the statistically self-similar case, asserts
that there is a positive probability of horizontal and vertical crossings if
$p$ is sufficiently close to 1.

######  Proposition 2.2.

There exist numbers $p_{H},p_{V}\in(0,1)$ such that:

if $0\leq p<p_{H}$ then $\theta_{H}(p)=0$ and if $p_{H}<p\leq 1$ then
$\theta_{H}(p)>0$,

if $0\leq p<p_{V}$ then $\theta_{V}(p)=0$ and if $p_{V}<p\leq 1$ then
$\theta_{V}(p)>0$.

The numbers $p_{H}$ and $p_{V}$ in Proposition 2.2 are termed critical
probabilities for H- and V-crossings respectively. Adding numerical details to
the proof, given in Section 3.1, gives upper bounds for the critical
probabilities that are extremely close close to 1 and which are far from
optimal. The method may be refined to get better values, but even for
percolation on a statistically self-similar set, the best-known bounds for the
critical probabilities are poor, see [4, 6, 19].

We will show that the horizontal and vertical critical probabilities are
equal, from which other properties of $F$ will follow.

######  Theorem 2.3.

Let $0\leq p\leq 1$. Then $\theta_{H}(p)>0$ if and only if $\theta_{V}(p)>0$.
In particular $p_{H}=p_{V}$.

Given this theorem we will write $p_{C}:=p_{H}=p_{V}$ for the common critical
value.

We will prove Theorem 2.3 in Section 3.2. We will then refine Proposition 2.2
to conclude that there is a strictly positive probability of both H- and
V-crossings at $p=p_{C}$ where $\theta_{H}(p)$ and $\theta_{V}(p)$ undergo a
jump phase transition.

######  Theorem 2.4.

The functions $\theta_{H}$ and $\theta_{V}$ are right-continuous and
increasing on $[0,1]$. If $0\leq p<p_{C}$ then $\theta_{H}(p)=\theta_{V}(p)=0$
and if $p_{C}\leq p\leq 1$ then $\theta_{H}(p),\theta_{V}(p)>0$.

Given Theorem 2.4, one can deduce more about $\theta_{H}(p)$ and
$\theta_{V}(p)$ as functions of $p$ as well as about the topological nature of
$F$ using methods analogous to those that have been developed for percolation
on statistically self-similar sets, in particular by Meester [14].

######  Theorem 2.5.

(i) If $1/mn<p<p_{C}$ then, conditional on $F\neq\emptyset$, the set $F$ is
uncountable and totally disconnected.

(ii) If $p_{C}\leq p<1$ then, conditional on $F\neq\emptyset$, the set $F$
includes countably many disjoint non-trivial connected components.

In Section 4 we indicate some other properties of $F$ and mention some open
questions.

##  3 Proofs

This section contains proofs of the results stated above.

###  3.1 Proofs of basic properties

Proof of Proposition 2.1 (i) Let $N_{p}(k)$ be the number of rectangles in
$E_{k}$ remaining at the $k$th stage of the construction of $F$. Then
$N_{p}(k)$ is a Galton-Watson branching process with offspring expectation
$pnm$. By standard branching process properties [1], if $pnm\leq 1$ then
almost surely the process becomes extinct so $F=\emptyset$, whereas if $pnm>1$
then $N_{p}(k)\not\to 0$ and indeed $N_{p}(k)\to\infty$, so that
$F\neq\emptyset$, with probability $t$, the least positive number such that
$t=\sum_{j=0}^{mn}\mathbb{P}(N_{p}(k)=j)t^{j}=\sum_{j=0}^{mn}\binom{mn}{j}p^{j}(1-p)^{mn-j}=(pt+1-p)^{mn}$.

(ii) These dimensions may be gleaned from more general results on dimensions
of statistically self-affine sets. For box-counting dimension see [2, Equation
(1.2)], [11, Theorem 4.1] or [18, Corollary 4.6], and for Hausdorff dimension
see [2, Equation (1.2)] or [18]. The calculations use that the projection of
$F$ onto the horizontal axis has box and Hausdorff dimensions
$\min\\{1,\log(pnm)/\log n\\}$ almost surely, subject to non-extinction. As
with many random constructions, the Assouad dimension is full, see [9, Theorem
3.2]. $\Box$

Proof of Proposition 2.2 We sketch this proof which is very similar to [4,
Section 2] or [5, Section 4]. The probability of a crossing increases with
$p$, and by part (i) is 0 if $0<p<1/m$ since then $\dim_{H}F<1$ and $F$ is
totally disconnected. Thus it is enough to show that there is some number
$p_{A}\in(0,1)$ such that if $p_{A}\leq p\leq 1$ there are, with positive
probability, both H- and V-crossings. Let $3\leq n<m$. We observe that if $R$
and $R^{\prime}$ are edge-adjacent level-$k$ rectangles in $E_{k}$, both
containing either $mn-1$ or $mn$ level-$(k+1)$ selected subrectangles in
$E_{k+1}$, then these subrectangles form a connected unit.

We say that a rectangle of $E_{k}$ is full if it contains $mn-1$ or $mn$
subrectangles of $E_{k+1}$. We say that a rectangle of $E_{k}$ is 2-full if it
contains $mn-1$ or $mn$ full subrectangles of $E_{k+1}$, and, inductively, a
rectangle of $E_{k}$ is $j$-full if it contains $mn-1$ or $mn$ subrectangles
of $E_{k+1}$ that are $(j-1)$-full. By the above observation, if
$E_{0}=[0,1]^{2}$ is $j$-full then opposite sides of the square $[0,1]^{2}$
are connected by a sequence of abutting rectangles in $E_{j}$.

Writing $p_{j}$ for the probability that $E_{0}$ is $j$-full, by using self-
affinity and the definition of $j$-full we obtain a polynomial recursion
$p_{j}=f_{p}(p_{j-1})$ where $f_{p}(t)=mnp^{mn-1}t^{mn-1}-(mn-1)p^{mn}t^{mn}$.
A calculation similar to [4, Section 2] shows that if $p$ is sufficiently
close to 1, say $p_{A}\leq p<1$, the polynomial $f_{p}$ has stable fixed point
$t_{p}\in(0,1)$ with $p_{j}=f_{p}^{j}(1)\to t_{p}$. Thus there is a positive
probability at least $t_{p}>0$ that $E_{0}$ is $j$-full for all
$j\in\mathbb{N}$, in which case there are both H- and V-crossings in $E_{j}$
for all $j$. As $E_{j}$ is a decreasing sequence of closed sets with
$F=\bigcap_{j=1}^{\infty}E_{j}$, we conclude that $F$ has both H- and
V-crossings with positive probability if $p\geq p_{A}$.

The same conclusion holds for the $2\times m$ random construction $(m\geq 3)$
with a suitable definition of ‘full’. $\Box$

###  3.2 Proof of Theorem 2.3

The following simple ideas are key to the proofs..

$\bullet$ If an event of positive probability is a finite or countable union
of sub-events, then at least one of the sub-events has positive probability.

$\bullet$ $F$ is statistically self-affine, that is for a level-$k$ rectangle
$R$, the random set $F_{k+1}\cap R$ has the same distribution as $\phi(F)$
(with the obvious translation to $R$) where $\phi$ is the affine map
$\phi(x,y)=(n^{-k}x,m^{-k}y)$. Similarly, $F\cap R$ and $F\cap R^{\prime}$
have identical distribution for level-$k$ rectangles $R$ and $R^{\prime}$.

$\bullet$ Limiting events such as crossings occur with positive probability in
$F$ if and only if it they occur with positive probability in $F_{k}$ for
some, and thus for all, $k\in\mathbb{N}$ (this follows since there is a
positive probability of selecting every level-$j$ rectangle for all $1\leq
j\leq k-1$).

$\bullet$ Harris’ inequality (or the FKG inequality), in the form that if
$A,B$ are increasing events then $\mathbb{P}(A\cap
B)\geq\mathbb{P}(A)\mathbb{P}(B)$, see [10, Section 2.2] or[4]. Recall that an
event $C$ is increasing if $I_{C}(\omega)\leq I_{C}(\omega^{\prime})$ whenever
$\omega\subseteq\omega^{\prime}$, where $I_{C}$ is the indicator function of
$C$, that is if an event occurs for a selection of rectangles it also occurs
for any larger selection. Many of the events we consider here, such as the
occurrence of crossings, are increasing.

We first note that $F$ does not contain line segments parallel to the axes.

######  Lemma 3.1.

Let $0\leq p<1$. Almost surely $F$ contains no horizontal and no vertical line
segments. The same is true for $F_{k}$ for all $k$.

###### Proof.

We show that $F$ contains no horizontal line segment, the vertical case is
similar.

First consider a line segment $S$ of the form

$[an^{-j},(a+1)n^{-j}]\times\\{bm^{-k}\\},\ \ j,k\in\mathbb{N},\ 0\leq a\leq
n^{j}-1,\ 0\leq b\leq m^{k}.$

Then for $q\geq j$ there are $n^{q-j}$ level-$q$ rectangles
$R_{i_{1},\ldots,i_{q}}$ that touch $S$ on each side. If $S\subset F$ then one
or other of every such abutting pair of level-$q$ rectangles must be selected,
and the probability of this is
$(1-(1-p)^{2})^{n^{q-j}}=(2p-p^{2})^{n^{q-j}}\to 0$ as $q\to\infty$. Thus
$\mathbb{P}(S\subset F)=0$ for all such $S$, and there are countably many
segments $S$ of this form.

Now let $j\geq 1$ and $0\leq a\leq m^{j}-1$; we consider rows of level-$q$
rectangles in the column $[an^{-j},(a+1)n^{-j}]\times[0,1]$. For $q\geq j$ the
row

$[cn^{-q},(c+1)n^{-q}]\times[bm^{-q},(b+1)m^{-q}],\ \ an^{q-j}\leq
c\leq(a+1)n^{q-j}-1$ (3.1)

contains $n^{q-j}$ level-$q$ rectangles each of which is selected
independently with probability $p$. The probability of all the rectangles in
this row being selected is $p^{n^{q-j}}$. There are $m^{q}$ such rows of the
form (3.1) with $0\leq b\leq m^{q}-1$ which are independent of each other, so
the probability of at least one such full row being selected is
$1-(1-p^{n^{q-j}})^{m^{q}}\to 0$ as $q\to\infty$, since
$m^{q}\log(1-p^{n^{q-j}})\sim-m^{q}p^{n^{q-j}}\to 0$.

A line segment $[an^{-j},(a+1)n^{-j}]\times\\{y\\}$ in $F$ where $y$ is not of
the form $bm^{-k}$ must lie in some row of selected rectangles of the form
(3.1) for every $q\geq j$, but almost surely there is no such sequence of rows
of selected rectangles.

Thus almost surely, $F$ contains no line segment of the form
$[an^{-j},(a+1)n^{-j}]\times\\{y\\}$ for each, and thus for all, $j$ and $a$.
Every horizontal line segment contains some interval of this form, so $F$
contains no horizontal line segments. The argument for $F_{k}$ is similar,
just considering $q\geq k$. ∎

We need the following intuitively obvious result on intersecting connected
components of a compact set.

######  Lemma 3.2.

Let $K$ be a non-degenerate compact convex subset of $\mathbb{R}^{2}$ with
perimeter curve $\partial K$. Let ${\bf x}_{1},{\bf x}_{2},{\bf x}_{3},{\bf
x}_{4}$ be (not-necesarily distinct) points in clockwise order on $\partial
K$. Let $F$ be a compact subset of $K$. If ${\bf x}_{1}$ and ${\bf x}_{3}$ are
connected in $F$ and ${\bf x}_{2}$ and ${\bf x}_{4}$ are connected in $F$ then
${\bf x}_{1},{\bf x}_{2},{\bf x}_{3},{\bf x}_{4}$ are all connected in $F$,
i.e. are all in the same connected component of $F$.

###### Proof.

If two of the ${\bf x}_{i}$ coincide then the conclusion follows since
connected components that intersect in a point must belong to a single
connected component.

Otherwise, suppose ${\bf x}_{1},{\bf x}_{3}$ are in a connected component $A$
of $F$ and ${\bf x}_{2},{\bf x}_{4}$ are in a connected component $B$. Then
$A,B$ are compact, so if they are disjoint they are separated by a positive
distance $d$. A compact set $E$ is connected if and only if for all
$\epsilon>0$ it is $\epsilon$-connected, that is for all ${\bf y}_{0},{\bf
y}\in E$ there is an $\epsilon$-chain of points ${\bf y}_{0},{\bf
y}_{1},\ldots,{\bf y}_{n}={\bf y}$ in $E$ such that $|{\bf y}_{i}-{\bf
y}_{i+1}|<\epsilon$ for all $0\leq i\leq n-1$, see [16]. Thus we may find a
$d/3$-chain in $A$ joining ${\bf x}_{1}$ and ${\bf x}_{3}$ and one in $B$
joining ${\bf x}_{2}$ and ${\bf x}_{4}$. The polygonal curves defined by these
chains must cross at some point ${\bf w}\in K$, so there are points in both
$A$ and $B$ within distance $d/3$ of ${\bf w}$, so $A$ and $B$ have separation
at most $2d/3$, a contradiction. Hence $A$ and $B$ are not disjoint, so are in
the same connected component of $F$. ∎

We now show that if there is a positive probability of $F$ crossing
horizontally a (perhaps narrow) column, we can find a ‘link’ component of $F$
of a particular form crossing the column, and then join up several such links
to obtain a crossing of $[0,1]^{2}$. This extends a technique used in [5].

For $k\in\mathbb{N}$ and $i\in\mathbb{Z}$ we write $I^{k}_{i}$ for the
interval

$I^{k}_{i}:=[im^{-k},(i+1)m^{-k}]\subset\mathbb{R}.$ (3.2)

For $0\leq x\leq 1$ let $L_{x}$ denote the vertical line-segment
$\\{x\\}\times[0,1]$.

######  Lemma 3.3.

Let $0<p\leq 1$ and let $q\geq 1$ be an integer. Suppose that there is
positive probability that $F$ includes a horizontal crossing of the column
$[0,n^{-q}]\times[0,1]$. Then there exist an integer $k>q$, intervals
$I^{k}_{i},I^{k}_{j},I^{k}_{i^{\prime}},I^{k}_{j^{\prime}}$ with
$|i-i^{\prime}|\geq 2$ and $|j-j^{\prime}|\geq 2$ , and integers $1\leq
a<\frac{1}{2}n^{k-q}<b<n^{k-q}$ such that with positive probability there is a
connected component $F_{0}$ of $F$ such that:

$(i)$ $F_{0}$ intersects the line segments $L:=\\{0\\}\times
I^{k}_{i},\,L^{\prime}:=\\{an^{-k}\\}\times
I^{k}_{i^{\prime}},\,R^{\prime}:=\\{bn^{-k}\\}\times
I^{k}_{j^{\prime}},\,R:=\\{n^{-q}\\}\times I^{k}_{j}$;

$(ii)$ $L$ is connected to $\L^{\prime}$ in
$F_{0}\cap([0,an^{-k}]\times[0,1])$;

$(iii)$ $R^{\prime}$ is connected to $R$ in
$F_{0}\cap([bn^{-k},n^{-q}]\times[0,1])$;

see Figure 3.

Figure 3: Line segments $L,L^{\prime},R^{\prime},R$ connected by a component
$F_{0}$ (represented here by a curve for illustrative convenience)

###### Proof.

If $F$ is a compact set that includes an H-crossing of
$[0,n^{-q}]\times[0,1]$, there is a connected component $F_{0}\subset
F\cap([0,n^{-q}]\times[0,1])$ that gives a H-crossing of
$[0,n^{-q}]\times[0,1]$.

Let $y\in[0,1]$ be such that the point $(0,y)\in F_{0}$. By Lemma 3.1 the
segment $[0,\frac{1}{2}n^{-q}]\times\\{y\\}$ is almost surely not contained in
the compact set $F$, so as the points $\\{an^{-k}\\}_{k,a}$ are dense in
$[0,1]$ there are almost surely $k,a\in\mathbb{N}$ with
$0<an^{-k}<\frac{1}{2}n^{-q}$ such that $(an^{-k},y)\notin F$. Since
$L_{an^{-k}}$ must intersect $F_{0}$ we may find $y^{\prime}\in[0,1]$ with
$(an^{-k},y^{\prime})\in F_{0}$ and $y^{\prime}\neq y$ such that
$(an^{-k},y^{\prime})$ is connected to $(0,y)$ in
$F_{0}\cap([0,an^{-k}]\times[0,1])$. There are intervals
$I^{k}_{i},I^{k}_{i^{\prime}}$ with $0\leq i,i^{\prime}\leq m^{k}-1$ and
$(0,y)\in\\{0\\}\times I^{k}_{i}$ and
$(an^{-k},y^{\prime})\in\\{an^{-k}\\}\times I^{k}_{i^{\prime}}$. If
$|i-i^{\prime}|<2$ then we may replace $k$ with a larger $k^{\prime}$ and find
smaller intervals $I^{k^{\prime}}_{i^{\prime\prime}}\ni y\text{ and
}I^{k^{\prime}}_{i^{\prime\prime\prime}}\ni y^{\prime}$ with
$|i^{\prime\prime}-i^{\prime\prime\prime}|\geq 2$ and
$a^{\prime}=an^{k^{\prime}-k}$ which satisfy ($ii$).

In the same way, working near the right-hand side of the rectangle
$[0,n^{-q}]\times[0,1]$, we may obtain intervals $I^{k}_{j}$ and
$I^{k}_{j^{\prime}}$ and an integer $b$ satisfying ($iii$). Finally, to ensure
a common value of $k$ in ($ii$) and ($iii$), we take the larger of the values
of $k$ obtained in ($ii$) and ($iii$) and reduce the lengths of the intervals
$I^{k}_{i},I^{k}_{i^{\prime}},I^{k}_{j},I^{k}_{j^{\prime}}$ as appropriate.

Thus if there is a positive probability that the random set $F$ provides an
H-crossing of $[0,n^{-q}]\times[0,1]$ then there are integers $k,a,b$ and
intervals $I^{k}_{i},I^{k}_{i^{\prime}},I^{k}_{j^{\prime}},I^{k}_{j}$ for
which a connected $F_{0}\subset F$ satisfies (i)-(iii). But there are
countably many possibilities for these integers and intervals, so at least one
combination occurs with positive probability. ∎

We proceed as in [5, Proof of Lemma 5.1] to link together columns of the form
considered in Lemma 3.3 to obtain a crossing of the unit square.

######  Proposition 3.4.

Let $0<p\leq 1$ and let $[r,s]\subset[0,1]$ be an interval. If there is a
positive probability that $F$ gives an H-crossing of the column
$[r,s]\times[0,1]$ then $\theta_{H}(p)>0$. Similarly, if there is a positive
probability that $F$ provides a V-crossing of the rectangle $[0,1]\times[r,s]$
then $\theta_{V}(p)>0$.

Figure 4: Constructing a horizontal crossing of $[0,1]^{2}$ given crossings of
vertical strips.

###### Proof.

Take $q\in\mathbb{N}$ such that $n^{-q}<\frac{1}{2}(r-s)$ and an integer
$0\leq u\leq n^{q}-1$ such that $[un^{-q},(u+1)n^{-q}]\subset[r,s]$. Then with
positive probability $[un^{-q},(u+1)n^{-q}]\times[0,1]$ is H-crossed by $F$
and thus by $F_{q}$. By statistical self-affinity the sets
$F_{q}\cap([un^{-q},(u+1)n^{-q}]\times[0,1])$ and
$F_{q}\cap([0,n^{-q}]\times[0,1])$ have the same distribution modulo a
horizontal shift of $un^{-q}$, so in particular there is positive probability
that $F_{q}$ includes an H-crossing of the rectangle
$K:=[0,n^{-q}]\times[0,1]$.

Lemma 3.3 provides an integer $k>q$ and specific segments
$L,L^{\prime},R^{\prime},R$ which with probability $p_{0}>0$, say, are
connected as in ($i$)-($iii$) in $F$ and thus in $F_{k}$ . Write
$C(K;L,L^{\prime},R^{\prime},R)$ for the event that $F_{k}$ has a component
that connects the vertical segments $L,L^{\prime},R^{\prime},R$ within the
rectangle $K$, connecting ${L}$ and $L^{\prime}$ within the vertical strip
with sides containing ${L}$ and $L^{\prime}$, and connecting $R^{\prime}$ and
$R$ within the vertical strip with sides containing $R^{\prime}$ and $R$. Let
$\rho$ denote the operation on $\mathbb{R}^{2}$ of reflection in the vertical
line $x=\frac{1}{2}(bn^{-k}+n^{-q})$, and let $\tau$ denote horizontal
translation by $(b-a+n^{k-q})n^{-k}$. These mapping have been set so that the
pair of right-hand intervals in the following chain of events align vertically
with the pair of left-hand intervals of the succeeding event.

We define a sequence of events:

$\begin{array}[]{ll}&C(K;L,L^{\prime},R^{\prime},R),\;C\big{(}\rho(K);\rho(R),\rho(R^{\prime}),\rho(L^{\prime}),\rho(L)\big{)},\\\
&C\big{(}\tau(K);\tau(L),\tau(L^{\prime}),\tau(R^{\prime}),\tau(R)\big{)},\;C\big{(}\tau\rho(K);\tau\rho(R),\tau\rho(R^{\prime}),\tau\rho(L^{\prime}),\tau\rho(L)\big{)},\\\
&C\big{(}\tau^{2}(K);\tau^{2}(L),\tau^{2}(L^{\prime}),\tau^{2}(R^{\prime}),\tau^{2}(R)\big{)},\;C\big{(}\tau^{2}\rho(K);\tau^{2}\rho(R),\tau^{2}\rho(R^{\prime}),\tau^{2}\rho(L^{\prime}),\tau^{2}\rho(L)\big{)},\\\
&C\big{(}\tau^{3}(K);\tau^{3}(L),\tau^{3}(L^{\prime}),\tau^{3}(R^{\prime}),\tau^{3}(R)\big{)},\ldots,\end{array}$
(3.3)

continuing until either $\tau^{\ell}(K)$ or $\tau^{\ell}\rho(K)$ overlaps the
right-hand edge of $[0,1]^{2}$ for some $\ell\in\mathbb{N}$ in which case we
ignore the level-$k$ rectangles outside the square.

Note that if consecutive events in this list occur then the intervals in both
events lie in the same component of $F_{k}$. For example, for $p\geq 0$,
$\tau^{p}(R^{\prime})$ and $\tau^{p}\rho(R)$ lie in a common vertical line as
do $\tau^{p}(R)$ and $\tau^{p}\rho(R^{\prime})$. Since $\tau^{p}(R)$ is joined
to $\tau^{p}(R^{\prime})$ and $\tau^{p}\rho(R)$ is joined to
$\tau^{p}\rho(R^{\prime})$ between these two vertical lines, Lemma 3.2 implies
that all four segments intersect a common component of $F_{k}$, and thus all
the segments listed in these two events are joined. In particular, if all the
events in this list occur, then there is a connected component of $F_{k}$ that
intersects $L$ and $\tau^{\ell}(R)$ or $\tau^{\ell}\rho(L)$ and so provides an
H-crossing of $[0,1]^{2}$.

But each event in the list has the same distribution as
$C(K;L,L^{\prime},R^{\prime},R)$, modulo a translation or reflection, so
occurs with probability $p_{0}$. These are all increasing events, so by
Harris’ inequality the probability of them all occurring is at least
$p_{0}^{2\ell+1}>0$, in which case there is a positive probability of an
H-crossing in $F_{k}$ and thus in $F$ (since there is positive probability of
all the rectangles up to level-$k$ all being selected).

The argument for vertical crossings is similar. ∎

Proof of Theorem 2.3 For $k\in\mathbb{N}$ and $0\leq b\leq m^{k}-1$ let
$A(k,b)$ be the event that $F$ includes a vertical crossing of the horizontal
strip $[0,1]\times[bm^{-k},(b+1)m^{-k}]$. Then

$\\{F\text{ has an H-crossing of
}[0,1]^{2}\\}\subset\bigcup_{k=1}^{\infty}\bigcup_{b=0}^{m^{k}-1}A(k,b)\cup\\{F\text{
contains a horizontal line}\\}.$

Thus if $\mathbb{P}\\{F\text{ has an H-crossing of }[0,1]^{2}\\}>0$ then
$\mathbb{P}(A(k,b))>0$ for some $k,b$, since $\mathbb{P}\\{F\text{ contains a
horizontal line}\\}=0$ by Lemma 3.1. Thus $F$ includes a V-crossing of
$[0,1]\times[bm^{-k},(b+1)m^{-k}]$ for some $k,b$, so $F$ has a V-crossing of
$[0,1]^{2}$ with positive probability by Proposition 3.4.

Hence if $\theta_{H}(p)>0$ then $\theta_{V}(p)>0$, with the reverse inequality
following using the ‘vertical’ part of Proposition 3.4. $\Box$

###  3.3 Proof of Theorems 2.4 and 2.5

Now we write $p_{C}:=p_{H}=p_{V}$.

For Theorem 2.4 we must show that $\theta_{H}(p_{C})>0$ and
$\theta_{V}(p_{C})>0$,which we do by adapting the proof in [5, Section 5]. It
seems awkward to show this directly, but rather we first consider crossings in
the random model starting with the rectangle $[0,1]\times[0,2]$ and proceeding
as before, with subdivision into a hierarchy of $n^{-k}\times m^{-k}$
rectangles for $k\geq 1$, with each rectangle selected with probability $p$.
In other words there are two vertically adjacent independent copies of the
original process in $[0,1]\times[0,1]$ and $[0,1]\times[1,2]$ with connection
across $[0,1]\times\\{1\\}$ permitted in the natural way. We write
$\tau_{H}(p)$ for the probability that there is a horizontal crossing of
$[0,1]\times[0,2]$. Similarly, we may consider percolation on the rectangle
$[0,2]\times[0,1]$, that is two horizontally adjacent copies of our original
model, and write $\tau_{V}(p)$ for the probability of a vertical crossing of
$[0,2]\times[0,1]$.

######  Lemma 3.5.

Let $0<p\leq 1$. The probabilities
$\theta_{H}(p),\theta_{V}(p),\tau_{H}(p),\tau_{V}(p)$ are either all $0$ or
all strictly positive.

###### Proof.

Clearly $\tau_{H}(p)=0$ implies $\theta_{H}(p)=0$. To show that if
$\tau_{H}(p)>0$ then $\theta_{H}(p)>0$ is a variant of [5, Proof of Lemma
5.1]. More specifically, this is similar to the construction of Lemma 3.3 but
starting with a positive probability H-crossing of the rectangle
$[0,1]\times[0,2]$ and extending a connected component horizontally in stages
by chaining together links as in the proof of Proposition 3.4 to obtain a
positive probability crossing in $F_{k}$ for some $k$, and thus in $F$, of
$[0,n]\times[0,2]$, considered as a $n\times 2$ array of independent copies of
the random process on $[0,1]\times[0,1]$. But the probability of an H-crossing
of $[0,n]\times[0,2]$ in $F$ is, by self-affinity, the same as the probability
of an H-crossing of $[0,1]\times[0,2m^{-1}]\subset[0,1]\times[0,1]$ in $F_{2}$
which is therefore positive giving a positive probability of an H-crossing in
$F$.

Thus $\tau_{H}(p)>0$ if and only if $\theta_{H}(p)>0$. Similarly
$\tau_{V}(p)>0$ if and only if $\theta_{V}(p)>0$. By Theorem 2.3
$\theta_{H}(p)>0$ if and only if $\theta_{V}(p)>0$, so the conclusion follows.
∎

The next lemma shows that for a given $p$, $\tau_{H}(p)$ and $\tau_{V}(p)$
cannot both be positive but very small.

######  Lemma 3.6.

Let $0<p\leq 1$. If $\theta_{H}(p)\geq\theta_{V}(p)$ then either
$\tau_{H}(p)=0$ or $\tau_{H}(p)\geq(4m)^{-n/(n-1)}$. If
$\theta_{V}(p)\geq\theta_{H}(p)$ then either $\tau_{V}(p)=0$ or
$\tau_{V}(p)\geq(4n)^{-m/(m-1)}$.

###### Proof.

Assume that $\theta_{H}(p)\geq\theta_{V}(p)$; the reverse case is similar. We
adapt the proof of [5, Theorem 5.4].

Consider one of the level-1 columns of $[0,1]\times[0,2]$ comprising $2m$ size
$m^{-1}\times n^{-1}$ rectangles, $Q_{1},\ldots,Q_{2m}$. If there is a
horizontal crossing of $[0,1]\times[0,2]$ and thus of this column, then at
least one of the following events occurs:

$V^{p}:=\bigcup_{i=1}^{2m}\\{\text{there exists a vertical crossing of
}Q_{i}\\}\text{ and }$ $H^{p}:=\bigcup_{i=1}^{2m-1}\\{\text{there exists a
horizontal crossing of }(Q_{i}\cup Q_{i+1})\\}.$

Using the statistical self-similarity between the process in $[0,1]^{2}$ and
each $Q_{i}$ and between that in $[0,1]\times[0,2]$ and each pair $Q_{i}\cup
Q_{i+1}$, allowing for the probability of these rectangles not being deleted
at level-1, and using that there are $n$ such columns, we relate the
probabilities:

$\displaystyle\tau_{H}(p)$ $\displaystyle\leq$
$\displaystyle(\mathbb{P}(V^{p}\cup
H^{p}))^{n}\leq\big{(}\mathbb{P}(V^{p})+\mathbb{P}(H^{p})\big{)}^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}2mp\,\theta_{V}(p)+(2m-1)\big{(}p^{2}\tau_{H}(p)+2p(1-p)\theta_{H}(p)\big{)}\Big{)}^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}2mp+(2m-1)\big{(}p^{2}+2p(1-p)\big{)}\Big{)}^{n}\tau_{H}(p)^{n}$
$\displaystyle\leq$ $\displaystyle\big{(}2m+(2m-1)\big{)}^{n}\tau_{H}(p)^{n}$
$\displaystyle\leq$ $\displaystyle(4m)^{n}\tau_{H}(p)^{n},$

using that $\theta_{V}(p)\leq\theta_{H}(p)\leq\tau_{H}(p)$. Thus
$\tau_{H}(p)=0$ or $\tau_{H}(p)\geq(4m)^{-n/(n-1)}$.

The case of $\theta_{V}(p)\geq\theta_{H}(p)$ is similar, considering crossings
of $[0,2]\times[0,1]$. ∎

Combining the previous two lemmas yields Theorem 2.4.

Proof of Theorem 2.4 First we note that $\theta_{H}(p)$ is increasing and
upper-semicontinuous and therefore right-continuous in $p$. To see this, the
probability $\theta^{k}_{H}(p)$ of a horizontal crossing of $[0,1]^{2}$ within
$E_{k}$ (see (2.1)) is a polynomial in $p$, since the probability of any given
configuration of rectangles at the first $k$ levels is a polynomial of the
form $p^{\alpha}(1-p)^{\beta}$, and summing the disjoint probabilities of the
configurations that include a crossing within $E_{k}$ gives a polynomial. Thus
$\theta^{k}_{H}(p)$ is increasing and continuous in $p$ for each $k$. As
$k\to\infty$, $\theta^{k}_{H}(p)\searrow\theta_{H}(p)$, recalling that there
is a crossing in $F$ if and only if there is a crossing in $E_{k}$ for all
$k$, so $\theta_{H}(p)$ is increasing and upper-semicontinuous, so right-
continuous, as the infimum of continuous functions.

Recall that $p_{C}$ is the common critical probability for $\theta_{H}(p)$ and
$\theta_{V}(p)$. There is a strictly decreasing sequence $p_{n}\searrow p_{C}$
either with $\theta_{H}(p_{n})\geq\theta_{V}(p_{n})$ for all $n$ or with
$\theta_{H}(p_{n})\leq\theta_{V}(p_{n})$ for all $n$, suppose the former, the
latter case is similar. By Lemma 3.6, $\tau_{H}(p_{n})\geq(4m)^{-n/(n-1)}$ for
all $n$, so since $\tau_{H}(p)$ is right-continuous,
$\tau_{H}(p_{C})\geq(4m)^{-n/(n-1)}$. By Lemma 3.5 $\theta_{H}(p_{C})>0$ and
$\theta_{V}(p_{C})>0$. $\Box$

We now establish Theorem 2.5 reflecting the nature of $F$ when $p<p_{C}$ and
when $p\geq p_{C}$.

Proof of Theorem 2.5

(i) Let $1/mn<p<p_{C}$ so conditional on $F\neq\emptyset$, $\dim_{H}F>0$ by
Proposition 2.1, so $F$ is uncountable.

Suppose there is a positive probability of a non-trivial connected component
$F_{0}$ of $F$. Then there are $a,k\in\mathbb{N}$ such that either $F$
includes an H-crossing of $[an^{-k},(a+1)n^{-k}]\times[0,1]$ or a V-crossing
of $[0,1]\times[an^{-k},(a+1)n^{-k}]$; by countable decomposition we may
select an orientation and specific $a$ and $k$ for which there is a positive
probability of this occurring. By Proposition 3.4 there is a positive
probability of either a V-crossing or an H-crossing of $[0,1]\times[0,1]$,
implying that $\theta_{H}(p),\theta_{V}(p)>0$, contradicting that $p<p_{C}$.

(ii) Let $p_{C}\leq p<1$. We call a rectangle $R$ in $E_{k}$ an island if
$R\cap F$ contains a non-trivial connected component and has an empty
perimeter band (i.e. the distance from $R\cap F$ to the perimeter of $R$ is
positive). There is a probability $p_{0}>0$ that $F$ is an island, since there
is a positive probability that none of the level-2 perimeter rectangles are
selected but some interior sub-rectangle is H-crossed by $F$, so by
statistical self-affinity there is a probability $p_{0}$ that each $R$ in
$E_{k}$ is an island. In particular each island in $E_{k}$ contains a non-
trivial connected component which is disjoint from all other level-$k$
rectangles.

Given $M\in\mathbb{N}$ we may proceed with the step by step construction of
$F$ until we stop at the first $k$ such that either $|E_{k}|\geq M$ or
$E_{k}=\emptyset$, the former will occur with probability at least $p_{N}$,
the probability of non-extinction of the construction. Conditioning on
$|E_{k}|\geq M$, the probability that at least $\frac{1}{2}p_{0}M$ of these
$R$ are islands is at least $1-4/(Mp_{0})$ using Chebychev’s inequality. By
taking $M$ large, it follows that the number of islands, and thus the number
of disjoint non-trivial connected components of $F$ is almost surely unbounded
conditional on non-extinction. $\Box$

##  4 Final remarks

Once it is established that the critical probabilities for H-crossings and
V-Crossings are equal, many other topological properties of the statistically
self-affine sets $F$ follow in a very similar way as for statistically self-
affine sets.

Path and arc connectivity (which are equivalent in $\mathbb{R}^{n}$) are
stronger conditions than topological connectivity. Nevertheless, the proof of
[14, Theorem 3.1] adapts to show that if $p_{C}\leq p\leq 1$ then with
positive probability $F$ is crossed both horizontally and vertically by a path
or arc; thus the critical probability for path/arc connectivity is the same as
for topological connectivity.

Similarly, adapting another proof of [14, Lemma 2.4] shows that almost surely
just finitely many disjoint connected components of $F$ cross $F$ horizontally
and finitely many cross vertically.

We end with some questions. Because in the $k$th level rectangular grid there
are far more rectangles in each column than in each row, one would expect that
the chance of an $V$-crossing would be much smaller than that of an
H-crossing. However, this does not seem obvious.

######  Question 4.1.

Is $\theta_{V}(p)\leq\theta_{H}(p)$ for all $m>n\geq 2$ and all $p_{C}\leq
p\leq 1$? Less strongly, is there a constant $c>0$ such that
$\theta_{V}(p)\leq c\,\theta_{H}(p)$ for all $p_{C}\leq p\leq 1$?

An affirmative answer to Question 4.1 would remove the need to consider two
cases in the proof of Theorem 2.4.

######  Question 4.2.

Are $\theta_{H}(p)$ and $\theta_{V}(p)$ continuous on the interval
$[p_{C},1]$?

This does not seem to be known even for the statistically self-similar
$m\times m$ case.

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