Non-Periodic Systems with
Continuous Diffraction Measures
The present state of mathematical diffraction theory for systems with continuous spectral components is reviewed and extended. We begin with a discussion of various characteristic examples with singular or absolutely continuous diffraction, and then continue with a more general exposition of a systematic approach via stationary stochastic point processes. Here, the intensity measure of the Palm measure takes the role of the autocorrelation measure in the traditional approach. We furthermore introduce a ‘Palm-type’ measure for general complex-valued random measures that are stationary and ergodic, and relate its intensity measure to the autocorrelation measure.
The (mathematical or kinematic) diffraction theory of systems in Euclidean space with pure point spectrum is rather well understood. Ultimately, this is due to the availability of Poisson’s summation formula and its generalisations to the setting of measures (or to tempered distributions); see [12, Sec. 9.2] for a systematic exposition. Beyond results on the spectral nature, this often also provides explicit formulas for the diffraction measure, such as in the cases of lattice-periodic systems and model sets. For these systems, there is also a well-understood connection with the Halmos–von Neumann theorem for the corresponding pure point dynamical spectrum; see [55, 16, 60, 18, 20] for details as well as  and references therein for general background.
As soon as one enters the realm of systems with continuous diffraction spectra (or at least with continuous spectral components), the situation changes drastically. As in the case of Schrödinger operator spectra , much less is known about the plethora of possibilities, and there rarely are explicit formulas for the diffraction measures of specific examples. Until recently, explicit results were restricted to simple systems of Bernoulli type (hence with disorder that leads to independent random variables) or to some paradigmatic examples in one dimension (and product systems built from them).
There has now been some progress towards explicitly computable examples in various directions [62, 4, 15]. In particular, both for singular and for absolutely continuous cases, constructive approaches have been more successful than previously anticipated; compare [12, Ch. 10]. Consequently, there is some hope that more systems can be understood in this way. This view is also supported by the recent progress in the understanding of the connection between the dynamical and the diffraction spectrum in this more general situation; see  and references therein. At the same time, such examples will improve our intuition about systems with continuous diffraction. Below, this will be reflected by several short sketches of characteristic examples (which are covered in more detail in ), before we embark on a more systematic setting via general point process theory. Our focus is on systems in , which is the primary situation to understand, particularly from the applications point of view. Extensions to more general locally compact Abelian groups are possible, but will not be discussed here.
2. Diffraction Measures — a Brief Reminder
Let be a locally finite (and possibly complex) measure on , which we primarily view as a linear functional on the space of continuous functions with compact support on , together with some mild extra conditions. In favourable cases, will be translation bounded. By the classic Riesz–Markov representation theorem, we may identify the measures defined by this approach with regular Borel measures; for a systematic exposition, we refer to [44, 16] as well as [12, Chs. 8 and 9] and references therein. Particularly important examples comprise the Dirac measure , defined by for , and measures of the form
which are known as Dirac combs, where is uniformly discrete. More generally, we will also consider objects of the form , which can be a measure for a general countable set , then under suitable conditions on the weight function . Such measures are referred to as weighted Dirac combs.
where denotes the volume averaged (or Eberlein) convolution. The latter is defined by
with the (open) ball of radius around the origin and . At this stage, we assume the existence of the limit. This will be discussed in more detail later.
If (as in many of our examples) is a Dirac comb with lattice support, also will be supported on the same lattice (or a subset of it). Concretely, if
with a bounded weight function say, one finds with the positive definite function being defined by
provided that all limits exist. In our exposition below, this existence will follow by suitable applications of Birkhoff’s ergodic theorem, applied to the dynamical system of the shift action on the orbit closure of the sequence or to a similar type of dynamical system; compare  for a more general setting. One benefit of this approach will emerge via the Herglotz–Bochner theorem .
The autocorrelation measure is positive definite (or of positive type) by construction, which means that for all . It is thus Fourier transformable , and the Fourier transform is a positive measure, called the diffraction measure of ; see  for the physics behind this notion, and  as well as [12, Ch. 9] for the mathematical theory. Within the framework of kinematic diffraction, it describes the outcome of a scattering experiment by quantifying how much intensity is scattered into a given volume of -space, and thus is the central object of our interest. By the Lebesgue decomposition theorem, there is a unique splitting
of the diffraction measure into its pure point part , its singular continuous part and its absolutely continuous part , with respect to Lebesgue measure . The pure point part comprises the ‘Bragg peaks’ (of which there are at most countably many, so is a sum over at most countably many Dirac measures with positive weights), while the absolutely continuous part corresponds to the diffuse ‘background’ scattering which is given by a locally integrable density relative to . The singular continuous part is whatever remains — if present, it is a measure that gives no weight to single points, but is still concentrated to an (uncountable) set of zero Lebesgue measure.
Measures which lead to a diffraction are called pure point diffractive; examples include lattice-periodic measures and measures based on model sets. These have been studied in detail in the context of diffraction of crystals and quasicrystals; see  for a recent review and [12, Chs. 8 and 9] for a systematic exposition. Here, we are concentrating on the other two spectral components, which may also carry important information on the (partial) order which is present in the underlying structure. Pure point spectra are discussed in detail in [23, 16, 17, 18, 12, 59, 57, 58]; for related spectral problems in the context of Schrödinger operators, we refer to .
3. Guiding Examples
As mentioned above, the understanding of systems with continuous diffraction components is less developed than that of pure point diffractive ones. Still, a better intuition will emerge from a sample of characteristic examples. It is the purpose of this section to provide some of them, while we refer to the literature for further ones [13, 22, 4, 9, 15, 12].
3.1. Thue–Morse Sequences
Let us begin with a classic example from the theory of substitution systems that leads to a singular continuous diffraction measure with rather different features in comparison with the Cantor measure, the latter being illustrated in Figure 1. Our example has a long history, which can be extracted from [79, 61, 47, 1]. We confine ourselves to a brief summary of the results, and refer to [7, 12] and references therein for proofs and details.
The classic Thue–Morse (TM) sequence can be defined via the one-sided fixed point (with ) of the primitive substitution rule
on the binary alphabet . The fixed point is the limit (in the obvious product topology) of the (suitably embedded) iteration sequence
and , for all ;
is (strongly) cube-free (and hence non-periodic).
Here, we define and identify with , also for the later calculations with Dirac combs. A two-sided sequence can be defined by
which is a fixed point of , because the seed is a legal word (it occurs in ) and . The (discrete) hull of the TM substitution is the closure of the orbit of under the shift action, which is a subset of and hence a compact space. The orbit of any of its members is dense in . We thus have a topological dynamical system that is minimal. When equipped with the standard Borel -algebra, the system admits a unique shift-invariant probability measure , so that the corresponding measure theoretic dynamical system is strictly ergodic [47, 68].
Any given is mapped to a signed Dirac comb via
The image of is a space of translation bounded measures that is compact in the vague topology. We inherit strict ergodicity via conjugacy, and thus obtain an autocorrelation of the form of Eq. (2.2) with coefficients as in Eq. (2.3). In fact, this autocorrelation does not depend on the choice of the element from , so that we may choose the fixed point from above for the concrete analysis. Due to the nature of , the coefficients can alternatively be expressed as
for , together with . It is clear that , and the scaling relations of lead to the recursions 
which are valid for all . In particular, the second relation, used with , implies , which can also be calculated directly.
Since is a positive definite function with , there is a unique probability measure on the unit circle (which we identify with the unit interval here) such that
which is a consequence of the Herglotz–Bochner theorem [50, Thm. I.7.6]. Since is supported on , the corresponding diffraction measure is -periodic, which follows from [3, Thm. 1]; see also [12, Sec. 10.3.2]. One then finds the relation
with the measure from Eq. (3.2), appropriately interpreted as a measure on and hence also on . Clearly, one also has . One can now analyse the spectral type of via that of the finite measure , where we follow ; see also [68, 20].
Defining , a two-step calculation with the recursion (3.1) establishes the inequality for all . This implies , wherefore Wiener’s criterion [79, 50], see also [12, Prop. 8.9], tells us that is a continuous measure, so that cannot have any pure point component. Note that the absence of the ‘trivial’ pure point component of on is due to the use of balanced weights, in the sense that and are equally frequent. Consequently, the average weight is zero, and the claim follows from [12, Prop. 9.2].
Let us now define the distribution function by for , which is a continuous function that defines a Riemann–Stieltjes measure [53, Ch. X], so that . The recursion relation for now implies  the two functional relations
we know that the coefficients must satisfy the same recursions (3.1) as , possibly with a different initial condition . The classic Riemann–Lebesgue lemma [50, Thm.I.2.8] states that . But this limit is only compatible with , because and for all , so that we must have . This means by the Fourier uniqueness theorem, wherefore and hence (neither of which is the zero measure) are purely singular continuous. The resulting distribution function is illustrated in Figure 2. Note that can consistently be extended to a continuous function on via for and then defines via in the Lebesgue–Stieltjes sense. The function can efficiently be calculated by means of the uniformly converging Volterra iteration
with . In contrast to the Devil’s staircase of Figure 1, the TM distribution function is strictly increasing, which means that there is no plateau (which would indicate a gap in the support of ); see [7, 12] and references therein for details and further properties of . So far, we have obtained the following result.
Let be any element of the Thue–Morse hull , the latter represented as a closed subshift of , and consider the corresponding Dirac comb . Then, its autocorrelation exists and is given by with being defined by Eq. (3.1) together with the initial condition .
The diffraction measure is , where is the purely singular continuous probability measure from Eq. (3.2). In particular, is purely singular continuous as well. ∎
To go one step further, Eq. (3.3) defines an iteration sequence of distribution functions for absolutely continuous measures that converges towards the TM measure in the vague topology. Writing , one finds
which results in an exact 2-to-1 surjection from the hull to . The latter is the hull of the period doubling substitution defined by
Viewed as topological dynamical systems, this means that is a factor of . Since both are strictly ergodic, this extends to the corresponding measure-theoretic dynamical systems. The period doubling sequence can be described as a regular model set with a -adic internal space [24, 23] and is thus pure point diffractive. This pairing also explains a phenomenon observed in , namely that the dynamical spectrum of the TM system is richer than its diffraction spectrum. By the dynamical (or von Neumann) spectrum, we mean the spectrum of the unitary operator induced by the shift on the Hilbert space , where is the unique shift-invariant probability measure on ; see  for more. Here, the pure point part of the dynamical spectrum is the ring , which is not even finitely generated (and only the ‘trivial’ part is detected by the diffraction measure of the TM system with general weights). In fact, our above measure from Theorem 3.1 represents the maximal spectral measure in the ortho-complement of the pure point sector [68, 20]. The missing pure point part, however, is fully recovered via the diffraction measure of ; see  for details and  for a general discussion of this phenomenon.
Various generalisations of this result are known by now. First of all, and perhaps not surprisiningly, this generalises to an entire family of bijective, binary substitutions . Moreover, extensions to higher dimensions are also possible, including the explicit nature of the resulting diffraction measure; compare [40, 11] and references therein.
3.2. Rudin–Shapiro Sequence
The (binary) Rudin–Shapiro (RS) chain is a bi-infinite deterministic sequence, with polynomial (in fact linear) complexity function and thus zero entropy. It can be described recursively as with , with initial conditions , and the recursion
which determines for all . The orbit closure of under the shift action is the (discrete) RS hull . Alternatively, one can start from a primitive substitution on a -letter alphabet (via , , and ) and define a quaternary hull, which then maps to the binary hull via a simple reduction to two letters (for instance via and ); compare [1, 68] or [12, Sec. 4.7.1] for details. The two hulls define topologically conjugate dynamical systems, with local derivation rules in both directions; see [12, Rem. 4.11].
The shift action on is strictly ergodic, so that one can define functions via
where all limits exist due to unique ergodicity (which is best formulated on the level of the -letter alphabet mentioned above). In particular, one finds and . The recursive structure of Eq. (3.6) now implies the validity of a closed set of recursive equations [8, 9], namely
which hold for all ; see [12, Sec. 10.2] for details. A careful inspection shows that the unique solution of this set of equations, with the initial conditions mentioned above, is together with for all . Hence, despite the deterministic nature of the RS sequence, the autocorrelation measure is simply given by , so that , where again denotes Lebesgue measure. Alternatively, the result also follows from the exposition in [68, 67].
Let be any element of the Rudin–Shapiro hull , and consider the corresponding Dirac comb . Then, its autocorrelation exists and is given by , with diffraction measure . ∎
As in the case of the TM sequence, the non-trivial pure point part of the dynamical spectrum (which is once again) is not ‘seen’ by the diffraction measure, while (with multiplicity ) represents once again the maximal spectral measure in the ortho-complement of the pure point sector. However, the missing pure point component can be recovered by a suitable factor system, the latter obtained via the block map defined by Eq. (3.4). The corresponding factor is represented by a limit-periodic substitution rule that is somewhat reminiscent of the paper folding sequence ; see [12, Sec. 10.2] for a complete discussion and  for the general connection between dynamical and diffraction spectra. The structure underlying the RS sequence can be generalised to higher-dimensional lattice substitutions in a rather systematic way; see  for details.
Let us begin this discussion by recalling the structure of the full Bernoulli shift from the viewpoint of kinematic diffraction. The classic coin tossing process leads to the Dirac comb
where the form an i.i.d. family of random variables, each taking values and with probabilities and , respectively. By an application of the strong law of large numbers (SLLN, see  for a favourable formulation), almost every realisation has the autocorrelation measure
and hence (via Fourier transform) the diffraction measure
Here, we have used the classic Poisson summation formula ; compare  and references therein, as well as [12, Sec. 9.2] for a formulation in the diffraction context. When , the diffraction boils down to . Here, the point part is extinct because the average scattering strength vanishes. For proofs, we refer the reader to [22, 4], while [51, 52] contain several important and non-trivial generalisations and extensions; see also  for important related material.
which is maximal for , with . It vanishes for the deterministic limiting cases . For the latter, we have , and consequently obtain the pure point diffraction measure , again via Poisson’s summation formula.
Now, the theory of random variables allows for an interpolation between deterministic (binary) sequences and coin tossing sequences as follows. If denotes a deterministic sequence (which we assume to be uniquely ergodic for simplicity), consider the random Dirac comb 
where is, as above, an i.i.d. family of random variables with values in and probabilities and . This ‘Bernoullisation’ of can be viewed as a ‘model of second thoughts’, where the sign of the weight at position is changed with probability ; compare [12, Sec. 11.2.2].
Let now be the Rudin–Shapiro sequence from above. By a (slightly more complicated) application of the SLLN, it can be shown  that the autocorrelation of the Dirac comb is then almost surely given by
irrespective of the value of the parameter . Recall that two measures with the same autocorrelation are called homometric; see [12, Sec. 9.6] for background. Our observation thus establishes the following classic result; see [8, 9, 12] for details.
The random Dirac combs of Equation (3.7) with real parameter values are almost surely homometric, with absolutely continuous diffraction measure , irrespective of the value of . In other words, the family is almost surely isospectral. ∎
This result shows that diffraction can be insensitive to entropy, because the family of Dirac combs of Eq. (3.7) continuously interpolates between the deterministic Rudin–Shapiro case with zero entropy and the completely random Bernoulli chain with maximal entropy . Clearly, the Bernoullisation procedure can be applied to other sequences as well, and can be generalised to higher dimensions. For further aspects of entropy versus diffraction, we refer to [8, 10, 19].
3.4. Random Dimers on the Line
Another instructive example  is based on certain dimer configurations on . To formulate it, we follow the exposition in  and partition into a close-packed arrangement of ‘dimers’ (pairs of neighbours), without gaps or overlaps. Clearly, there are just two possibilities to do so, because the position of the first dimer fixes that of all others. Next, decorate each dimer randomly with either or , with equal probability. This results in patches such as
where the dimer boxes are indicated by brackets. The set of all decorated sequences defined in this way is given by
where . Note that is empty precisely for the two periodic sequences that are defined by for . Clearly, is closed and hence compact.
Let and consider the corresponding signed Dirac comb on with binary weights . One can then show (again via the SLLN) that the corresponding autocorrelation almost surely exists and is given by 
The corresponding diffraction measure is then
which is again purely absolutely continuous. Here, the (smooth) Radon–Nikodym density relative to is written as a function of . Note that the diffraction measure for general weights and is given by
with as in Eq. (3.9). In particular, the measure shows only the ‘trivial’ pure point diffraction contribution that arises as the consequence of being the support of the weighted measure under consideration. The same phenomenon also occurs for general (non-balanced) TM and RS sequences; compare [12, Rems. 10.3 and 10.5].
On first sight, the system looks disordered, with entropy . This seems (qualitatively) reflected by the diffraction. However, the system also defines a measure-theoretic dynamical system under the action of , as generated by the shift. As such, it has a dynamical spectrum that does contain a pure point part, with eigenvalues and ; we refer to  for general background on this concept, and to  for the actual calculation of the eigenfunctions. The extension to a (continuous) dynamical system under the general translation action of is done via suspension; see [28, Ch. 11.1] (where the suspension is called a special flow) or  for general background.
This finding suggests that some degree of order must be present that is neither visible from the entropy calculation nor from the diffraction measure alone. Indeed, in analogy with the situation of the TM and the RS sequence, one can define a factor of the system by a sliding block map defined by . It maps globally 2:1 onto
The suspension (for the action of ) is defined as above. The mapping extends accordingly.
The autocorrelation and diffraction measures of the signed Dirac comb for an element are almost surely given by
The diffraction of the factor system uncovers the ‘hidden’ pure point part of the dynamical spectrum, which was absent in the purely absolutely continuous diffraction of the signed Dirac comb with . In summary, we have the following situation [25, 20].
The diffraction measure of the close-packed dimer system with balanced weights is purely absolutely continuous and given by Eq. (3.9), which holds almost surely relative to the natural invariant measure of the system.
The dynamical spectrum of the continuous close-packed dimer system under the translation action of contains the pure point part together with a countable Lebesgue spectrum.
The non-trivial part of the dynamical point spectrum is not reflected by the diffraction spectrum of , but can be recovered via the diffraction spectrum of a suitable factor, such as . ∎
As in the case of the Thue–Morse system, where the missing pure point part of the dynamical spectrum is recovered by the diffraction measure of the period doubling factor, we thus see that and how we can recover the missing eigenvalue via a generalised -point function. This observation can be extended to symbolic systems over finite alphabets and also to uniquely ergodic Delone dynamical systems of finite local complexity; see  for details.
3.5. Ledrappier’s Shift Space
For a long time, people had expected that higher dimensions are perhaps more difficult, but not substantially different. This turned out to be a false premise though, as can be seen from the now classic monograph .
In our present context, we pick one characteristic example, the system due to Ledrappier , to demonstrate a new phenomenon. We follow the brief exposition in  and consider a specific subset of the full shift space , defined by
where and denote the standard Euclidean basis vectors in the plane. On top of being a closed subshift, is also an Abelian group (here written multiplicatively), which then comes with a unique, normalised Haar measure. The latter is also shift invariant, and the most natural measure to be considered in our context; see also the reformulation in terms of Gibbs (or equilibrium) measures in .
The system is interesting because the number of patches of a given radius (up to translations) grows exponentially in the radius rather than in the area of the patch. This phenomenon is called entropy of rank , and indicates a new class of systems in higher dimensions. More precisely, along any lattice direction of , the linear subsystems essentially behave like one-dimensional Bernoulli chains. It is thus not too surprising that the diffraction measure satisfies the following theorem, though its proof  has to take care of the special directions connected with the defining relations of .
If is an element of the Ledrappier subshift of Eq. (3.10), the corresponding weighted Dirac comb has diffraction measure , which holds almost surely relative to the Haar measure of . ∎
So, the Ledrappier system is homometric to the (full) Bernoulli shift on , which means that an element of either system almost surely has diffraction measure . As mentioned before, via a suitable product of two Rudin–Shapiro chains, also a deterministic system with diffraction exists. This clearly demonstrates the insensitivity of pair correlations to the (entropic) type of order or disorder in the underlying system; see also . Due to the defining relation in Eq. (3.10), it is clear that certain three-point correlations in the Ledrappier system cannot vanish, and thus make it distinguishable from the Bernoulli shift.
Although correlation functions of third order can resolve the situation in this case (and in many other examples as well [32, 57]), one can consider other dynamical systems (such as the ()-shift ) that share almost all correlation functions with the Bernoulli shift on . This is a clear indication that our present understanding of ‘order’ is incomplete, and that we still lack a good set of tools for the detection and classification of order. For a recent alternative based on direct space statistics, we refer to .
3.6. Random Matrix Ensembles
Another interesting class of random point sets derives from the (scaled) eigenvalue distribution of certain random matrix ensembles; see  and references therein. The global eigenvalue distribution of random orthogonal, unitary or symplectic matrix ensembles is known to asymptotically follow the classic semi-circle law. More precisely, this law describes the eigenvalue distribution of the underlying ensembles of symmetric, Hermitian or (symplectically) self-dual matrices with Gaussian distributed entries. The corresponding random matrix ensembles are called GOE, GUE and GSE, with attached -parameters , and , respectively. They permit an interpretation as a Coulomb gas, where is the power in the central potential; see [2, 63] for general background and [34, 38] for the results that are relevant here.
For matrices of dimension , the semi-circle has radius and area . Note that, in comparison with , we have rescaled the density by a factor , so that we really have a semi-circle (and not a semi-ellipse). To study the local eigenvalue distribution for diffraction, we rescale the central region (between , say) by . This leads, in the limit as , to an ensemble of point sets on the line that can be interpreted as a stationary, ergodic point process of intensity ; for , see [2, Ch. 4.2] and references therein for details. Since the underlying process is simple (meaning that, almost surely, no point is occupied twice), almost all realisations are point sets of density .
It is possible to calculate the autocorrelation of these processes, on the basis of Dyson’s correlation functions . Though these functions originally apply to the circular ensembles, they have been adapted to the other ensembles by Mehta . For all three ensembles mentioned above, this leads to an autocorrelation of the form
A similar approach is possible on the basis of the eigenvalues of general complex random matrices. This leads to the ensemble studied by Ginibre , which is also discussed in . One common feature of the resulting point sets is the effectively repulsive behaviour of the points, which leads to the ‘dip’ around for . For the two systems mentioned in this section, we omit the formulation of the full results and refer the reader to  for details. Further developments around determinantal and related point processes are described in reference .
4. The Renewal Process
A large and interesting class of processes in one dimension can be described as a renewal process [37, 4, 14]. Here, one starts from a probability measure on (the positive real line) and considers a machine that moves at constant speed along the real line and drops a point on the line with a waiting time that is distributed according to . Whenever this happens, the internal clock is reset and the process resumes. Let us (for simplicity) assume that both the velocity of the machine and the expectation value of are , so that we end up with realisations that are, almost surely, point sets in of density (after we let the starting point of the machine move to , say).
Clearly, the resulting process is stationary and can thus be analysed by considering all realisations which contain the origin. Moreover, there is a clear (distributional) symmetry around the origin, so that we can determine the corresponding autocorrelation of almost all realisations from studying what happens to the right of . Indeed, if we want to know the frequency per unit length of the occurrence of two points at distance (or the corresponding density), we need to sum the contributions that is the first point after , the second point, the third, and so on. In other words, we almost surely obtain the autocorrelation
with , where the proper convergence of the sum of iterated convolutions follows from [4, Lemma 4] or from [12, Sec. 11.3]. Note that the point measure at simply reflects the fact that the almost sure density of the resulting point set is . Indeed, is a translation bounded positive measure, and satisfies the renewal relations (compare [37, Ch. XI.9] or [4, Prop. 1] for a proof)
where is a uniformly continuous and bounded function on . The second equation emerges from the first by Fourier transform, but has been rearranged to highlight the relevance of the set of singularities. In this setting, the measure of Eq. (4.1) is both positive and positive definite.
Based on the structure of the support of the underlying probability measure , one can determine the diffraction of the renewal process explicitly. To do so for a probability measure on with mean , we assume the existence of a moment of of order for some ; we refer to  for details on this condition. The diffraction measure of the point set realisations of the stationary renewal process based on almost surely is of the form
where is a locally integrable function on that is continuous almost everywhere. The pure point part is trivial, meaning , unless the support of is contained in a lattice. The details are stated below in Theorem 4.1. Proofs of these claims as well as further results can be found in [4, 14, 12].
The renewal process is a versatile method to produce interesting point sets on the line. These include random tilings with finitely many intervals (which are Delone sets) as well as the homogeneous Poisson process on the line (where is the exponential distribution with mean ); see [4, Sec. 3] for explicit examples and applications. In particular, if one employs a suitably normalised version of the Gamma distribution, one can formulate a one-parameter family of renewal processes that continuously interpolates between the Poisson process (total positional randomness) and the lattice (perfect periodic order); compare [4, Ex. 3] for more. The general result reads as follows.
Let be a probability measure on with mean , and assume that a moment of of order exists for some . Then, the point sets obtained from the stationary renewal process based on almost surely have a diffraction measure of the form
where is a locally integrable function on that is continuous except for at most countably many points namely those of the set . On , the function is given by
Moreover, the pure point part is
where is the coarsest lattice that contains . ∎
In one dimension, the renewal process allows an efficient derivation of the diffraction of random tilings, which we briefly summarise now.
5. Random Tilings
The deterministic Fibonacci chain can be defined by the primitive substitution rule , , which defines a strictly ergodic (discrete) hull. When and are replaced by intervals of length and , respectively, the left endpoints of the intervals define a model set (or cut and project set). The corresponding Dirac comb leads to the pure point diffraction measure
with intensities . Here, is the density of the point set, and denotes the algebraic conjugate of , which is defined on the field by and acts as the -map for the underlying model set description. In particular, the diffraction is the same for all Dirac combs of the Fibonacci hull; see [12, Sec. 9.4.1] and references therein for details. An illustration is shown in the upper panel of Figure 4.
The corresponding random tiling ensemble consists of all tilings of the real line by the two types of intervals. For a direct comparison, it makes more sense to only consider those tilings with the same relative frequency of interval types, which means frequencies and for the long and the short interval, respectively.
with the Radon–Nikodym density
Except for the trivial Bragg peak at , the diffraction measure is thus absolutely continuous. Still, the resemblance between this function and the diffraction of the perfect Fibonacci chain is remarkable, as can be seen from Figure 4.
The situation in dimensions is less favourable from a mathematical perspective, although one has a rather clear intuition of what one should expect [43, 70], based on solid scaling arguments. In dimensions , a mixed spectrum with pure point and absolutely continuous components is conjectured, while is the critical dimension in the sense that random tilings with non-crystallographic symmetries should display a singular continuous component; see [12, Sec. 11.6.2] for an example.
Unfortunately, only few results have been proved so far. Among them are a rigorous treatment of planar random tiling ensembles with crystallographic symmetries (such as the lozenge tiling and several relatives, see [13, 45, 46]), a group-theoretic approach to one of the random tiling hypotheses [69, 70] and a treatment of dense Dirac combs with pure point diffraction [71, 59] that is needed to understand the pure point part of the random tiling diffraction in dimensions . The remaining questions are still open, though there is little doubt that the original analysis from  is essentially correct.
6. Stochastic Point Processes and the Palm Measure
In this section, we take the viewpoint of a general shift-invariant random measure and relate its realisation-wise diffraction to its second moment measure. As such, this section is a complex-valued extension of [4, Sec. 5].
Let be a locally finite complex-valued measure on (which means that and are both locally finite signed measures). A short calculation reveals that, for of the form with real-valued and , the measure can consistently be defined via
where with . In particular, note that
hold as expected. The point here is that, after having dealt with the case of real (or signed) measures, the extension to complex measures is canonical and consistent.
where is the total variation measure of . This means that is the smallest non-negative measure such that for any bounded and measurable , where ; compare [12, Sec. 8.5.1] and references therein.
Let denote the -vector space of all locally finite, complex-valued measures on , so means for any bounded Borel set . A sequence converges vaguely to if as for all . The space is closed in the topology of vague convergence of measures (in fact, is a Polish space with this topology). We let denote the -algebra of Borel sets of . The latter can be described as the -algebra of subsets of generated by the requirement that, for all bounded Borel sets , the mapping is measurable.
For each , let denote the translation operator on , as defined by the mapping . Clearly, , and the inverse of is given by . For functions on , the corresponding translation action is defined via , so that . Similarly, for , let be the image measure under the translation, so that for any measurable subset , and for functions. This means that there is a translation action of on . Finally, we also have a translation action on , the probability measures on , via for any and . A set is called invariant (under translations) if for all .
A (complex-valued) random measure is a random variable (defined on some probability space ) with values in , which formally means that is an -measurable function. Its distribution is then , i.e. the image measure of under . We will follow the usual practice in probability theory and not make the underlying probability space explicit (a canonical choice can in many cases simply be and ). We will also usually suppress the dependence of on in the notation. Integrals over w.r.t. the probability measure will be denoted by , the expectation value.
is called stationary if its distribution satisfies for all . A stationary random measure is called ergodic if the shift-invariant -algebra is trivial, which means that any invariant has probability or (more generally, one requires whenever for all ; compare [30, Def. 10.3.I and Prop. 10.3.III]).
In what follows, we generally assume that
which means that there is a decomposition where both and are signed, real-valued, stationary, ergodic random measures on . To verify the last statement note that, since for any bounded measurable , is measurable, also and are measurable as functions of . Consider any shift-invariant measurable ( denotes the locally finite signed measures on ), then is shift invariant and measurable as well, so , and analogously for . We further assume that is locally square integrable in the sense that
where and denote the total variation measures of and , respectively.
In analogy with the real-valued case in [4, Sec. 5.2], we define , the second moment measure of , via
hence, for ,
By the shift invariance of the distribution of , we have
for all , and hence we can factor out this symmetry to obtain the reduced second moment measure . The latter is a locally finite complex-valued measure that is characterised by
for . By the shift invariance of Lebesgue measure on , we equivalently have
To prove the existence of , one can decompose