Generalized Random Energy Model

Generalized Random Energy Model at Complex Temperatures

Abstract.

Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature . We compute the limiting log-partition function and describe the fluctuations of the partition function. For the GREM with levels, in total, there are phases, each of which can symbolically be encoded as with such that . In phase , the first levels (counting from the root of the GREM tree) are in the glassy phase (G), the next levels are dominated by fluctuations (F), and the last levels are dominated by the expectation (E). Only the phases of the form intersect the real axis. We describe the limiting distribution of the zeros of the partition function in the complex plane (= Fisher zeros). It turns out that the complex zeros densely touch the positive real axis at points at which the GREM is known to undergo phase transitions. Our results confirm rigorously and considerably extend the replica-method predictions from the physics literature.

Key words and phrases:
Generalized Random Energy Model, complex inverse temperature, spin glasses, partition function, Lee–Yang program, Fisher zeros, extreme values, Poisson cascade zeta function, lines of zeros
2010 Mathematics Subject Classification:
Primary: 82B44; secondary: 60G50, 60G70, 60E07, 60K35, 60F05, 60F17, 30C15, 60G55, 60G52

1. Introduction and definition of the model

1.1. Introduction

The study of phase transitions is one of the central topics in statistical physics. Phase transitions are usually defined as the values of physical parameters (for example, the inverse temperature ) at which the limiting log-partition function (or equivalently the free energy) is not real analytic (= non-analytic in any neighborhood of the phase transition point). However, at any finite system size, the log-partition function is real analytic. In order to explain why the infinite system log-partition function looses analyticity (while the finite system log-partition function does not), Lee and Yang [48, 29] suggested to look at complex values of the inverse temperature . At complex temperatures, the partition function may have zeros and hence, the log-partition function has singularities, even for finite system sizes. If in the infinite system limit these singularities accumulate around the real axis at some , then the limiting log-partition function may loose analyticity at , even though itself is never a point of singularity of the log-partition function. Thus, the approach of Lee and Yang relates phase transitions to the distribution of complex zeros of the partition function. The study of the complex zeros of the partition function is usually referred to as the Lee–Yang program; see for example [3, 4], where a large class of lattice spin models is considered from this point of view.

The aim of the present work is to study a special model of spin glass, the Generalized Random Energy Model (GREM) within the Lee–Yang program. The simplest model of a spin glass is the Random Energy Model (REM) introduced by Derrida [12, 13]. In this model, the energies of the system are assumed to be independent Gaussian random variables. The behavior of the REM at real inverse temperature is well understood; see Bovier et al. [10] and Bovier [5, Chapter 9]. For the REM at complex inverse temperature, Derrida [15] derived the limiting free energy, obtained the phase diagram and computed the limiting distribution of complex zeros of the partition function. The present authors refined Derrida’s results and provided rigorous proofs in [25].

Although the REM contains some of the physics of the spin glasses, e.g., it displays the freezing phenomenon, the REM does not exhibit such phenomena as multiple freezing transitions and chaos which are observed, e.g., in the celebrated Sherrington–Kirkpatrick (SK) model of a spin glass. In order to obtain a solvable model with multiple freezing transitions, Derrida introduced the Generalized Random Energy Model (GREM); see [14, 16, 17]. Rigorous results on the GREM at real inverse temperatures were obtained by Capocaccia et al. [11] and in a series of works by Bovier and Kurkova [7, 8, 6]. For a review of these results, we refer to Bovier and Kurkova [9] and Bovier [5, Chapter 10]. We note in passing that the recent progress in rigorous understanding of the SK model draws heavily on the analysis of the GREM, see [34] for a review.

In the theoretical physics literature, there is a strong interest in studying spin glass models at complex temperatures. Besides the Lee–Yang program, the motivation comes here from quantum physics and concretely from the studies of interference in inhomogeneous media. See, e.g., the recent works of Takahashi and Obuchi [33, 45, 46], Saakian [40, 41], Dobrinevski et al. [19]. In particular, Takahashi [45], developed a complex version of the (non-rigorous) replica method and used it to identify the phase diagram of the GREM.

As for the rigorous works, beyond the uncorrelated case of the REM, to our knowledge, only two models of disordered systems with correlated complex random energies have been studied to some extent: the Branching Random Walk and the Gaussian Multiplicative Chaos. See Derrida et al. [18] and the recent works of Lacoin et al. [27] and Madaule et al. [30, 31]. Both models have correlations of logarithmic type and their complex-plane phase diagrams are quite similar to that of the REM (see Section 2.10 for more details).

The GREM seems to be a natural candidate to be tackled next from the Lee–Yang viewpoint. On the one hand, as we show below, the complex GREM is a rather tractable model even at the level of fluctuations, and, on the other hand, it exhibits multiple freezing phase transitions and has a much richer phase diagram than that of the REM.

The main results of this paper can be summarized as follows:

1. we compute the limiting log-partition function ;

2. we describe the global limiting distribution of complex zeros of ;

3. we identify the limiting fluctuations of ;

4. we prove functional limit theorems for in a suitably rescaled neighborhood of a fixed ;

5. we describe the local limiting distribution of complex zeros of in a suitably rescaled neighborhood of a fixed .

These results give the complete phase diagram of the GREM; see Figures 1 and 4. Our results confirm the replica-method predictions of Takahashi [45] and extend these considerably. We also indicate how to pass to the limit of continuous hierarchies (Continuous Random Energy Model, CREM), see Section 2.10, which allows us to compare our results with the ones on on the Branching Random Walk [30] and the Gaussian Multiplicative Chaos [27, 31]. We hope that our results shed more light on the complex plane phase diagrams and on fluctuations in strongly correlated random energy models.

1.2. Notation: Definition of the GREM

We start by introducing the notation which will be used throughout the paper. Fix the following parameters:

1. the number of levels ;

2. the variances of the levels (energetic parameters);

3. the branching exponents (entropic parameters).

We also fix sequences of natural numbers (called the branching numbers) such that for every ,

 (1.1) limn→∞Nn,jαnj=1.

The reader may simply take . Consider a rooted tree, denoted by , which is constructed in the following way. The root of the tree is located at level and is connected by edges to vertices (descendants) at level . Any vertex at level is connected to vertices at level , and so on. Finally, any vertex at level is connected to terminal vertices (leaf nodes) which have no descendants. We label the edges of the tree by levels so that the edges issuing from the root are at level , whereas the leaf edges of the tree are at level . The set of paths in connecting the root to the terminal vertices is denoted by

 (1.2) Sn={ε=(ε1,…,εd)∈Nd:1≤ε1≤Nn,1,…,1≤εd≤Nn,d}.

The total number of elements in and its growth exponent are given by

 (1.3) Nn:=#Sn=Nn,1⋅Nn,2⋅…⋅Nn,d∼αn,α:=α1⋅α2⋅…⋅αd.

Consider independent real standard normal random variables attached to the edges of the tree and denoted by

 (1.4) {ξε1…εj:1≤j≤d,1≤ε1≤Nn,1,…,1≤εj≤Nn,j}.

Define a zero-mean Gaussian random field by

Note that the variance of this random field is constant:

In the literature on the GREM, one usually assumes that the total number of energies in is (so that ) and that the variance is . Since we will often use induction over the number of levels of the GREM, it is more convenient to us to consider the general case omitting these assumptions.

Let us write the complex inverse temperature in the form

 β=σ+iτ∈C,σ=Reβ∈R,τ=Imβ∈R.

The partition function of the Generalized Random Energy Model at inverse temperature is defined by

 (1.7) Zn(β)=∑ε∈Sneβ√nXε.

Define the critical inverse temperatures

 (1.8) σj=√2logαjaj∈R,1≤j≤d.

To make the notation consistent, we make the convention and . Throughout the whole paper, we assume that

 (1.9) σ1<…<σd.

Geometrically, this condition means that the broken line joining the points

 (a1+…+aj,logα1+…+logαj),0≤j≤d,

is strictly concave. If (1.9) is not satisfied, one has to coarse grain the GREM levels by replacing the above broken line by its concave hull; see [7] for details in the real case. If (1.9) does not hold, there are less phase transition temperatures than . In order to avoid complicated notation, we assume (1.9).

Often, we can restrict ourselves to the quarter-plane and because of the straightforward distributional equalities

 (1.10) {Zn(−β):β∈C} \lx@stackreld={Zn(β):β∈C}, (1.11) {Zn(¯β):β∈C} \lx@stackreld={¯¯¯¯¯¯¯¯¯¯¯¯¯¯Zn(β):β∈C}.

1.3. Notation: Spaces and modes of convergence

In this section, we briefly recall several notions of convergence which will be frequently used below. For more information, we refer to the classical books [2] and [26]. The reader may skip this section and return to it when necessary.

Let be a locally compact metric space with metric . If not stated otherwise, all measures on are defined on the Borel -algebra generated by the metric .

Space of Radon measures.

A Radon measure on is a measure on having the property that for every compact set . Let be the set of all Radon measures on . A sequence of Radon measures converges vaguely to a Radon measure if for every continuous compactly supported function we have . Endowed with the topology of vague convergence, becomes a Polish space. A random measure on is a random variable defined on some probability space and taking values in .

Space of integer-valued Radon measures.

Let be the subset of consisting of all measures such that for every compact set . Measures with this property are called integer-valued. Every measure can be represented as , where is at most countable collection of points in having no accumulation points in . Here, is the Dirac delta-measure at . It is well-known that is a closed subset of . We endow with the induced vague topology. A point process on is a random variable defined on some probability space and taking values in .

Space of continuous functions.

Recall that is a locally compact metric space with metric . Let be the space of all (not necessarily bounded) continuous complex-valued functions on . A sequence of continuous functions on converges locally uniformly if it converges uniformly on every compact set . Endowed with the topology of locally uniform convergence, the space becomes a Polish space. A random continuous function on is a random variable defined on some probability space and taking values in .

If is an open subset of , let be the set of all complex-valued functions which are analytic on . Note that is a closed linear subspace of . We endow with the topology of locally uniform convergence induced from . A random analytic function on is a random variable defined on some probability space and taking values in .

Weak convergence.

Let be a metric space. A sequence of random elements taking values in converges weakly to a random element with values in if for every continuous, bounded function , we have . In the case when is , , , or , we speak of weak convergence of random measures, point processes, random continuous functions, or random analytic functions, respectively.

Zeros of analytic functions.

For an analytic function which is defined on some domain (=connected open set) and does not vanish identically, we denote by an integer-valued Radon measure on which counts the zeros of in according to their multiplicities.

Real and complex Gaussian distribution.

The real Gaussian distribution with mean zero and variance has density

 φR(t)=1√2πθe−t22θ2,t∈R,

w.r.t. the Lebesgue measure on . The complex Gaussian distribution with mean zero and variance has density

 φC(t)=1πθ2e−|t|2θ2,t∈C,

w.r.t. the Lebesgue measure on . Note that iff , where are independent. A zero mean real or complex Gaussian distribution is called standard if .

Throughout the paper, denote positive constants whose values may change from line to line. Let . We write if .

2. Statement of results

2.1. Limiting log-partition function

In this section, we state a formula for the limiting log-partition function of the GREM. To understand this formula heuristically, imagine a GREM with levels as a “superposition” of independent copies of the REM. (Note that the random field which generates the partition function of the GREM, cf. (1.7), has strong correlations.) Namely, with every level of the GREM we can associate a REM whose partition function is given by

 (2.1) Z(k)n(β)=Nn,k∑j=1eβ√nakη(k)j,1≤k≤d,

where are independent real standard normal random variables. The complex plane phase diagram of the REM has been described by Derrida [15]; see also [25]. There are three phases, see Figure 2, which we will denote by

1. (expectation dominated phase),

2. (fluctuations dominated phase),

3. (“glassy phase” = extreme values dominated phase).

Concretely, the phases are given by

 (2.2) Gk ={β∈C:2|σ|>σk,|σ|+|τ|>σk}, (2.3) Fk ={β∈C:2|σ|<σk,2(σ2+τ2)>σ2k}, (2.4) Ek =C∖¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Gk∪Fk,

where is the closure of the set . The phases and intersect the real axis, while the phase is special for the complex case. By definition, the sets , , are open.

Derrida [15], see also [25] for a rigorous proof, computed the limiting log-partition function of the REM at complex . Namely, for the log-partition function of the REM corresponding to the -th level of the GREM,

 (2.5) pk(β)=limn→∞1nlog|Z(k)n(β)|,

Derrida’s formula takes the form

 (2.6) pk(β)=⎧⎪ ⎪⎨⎪ ⎪⎩|σ|√2aklogαk,if β∈¯Gk,12logαk+akσ2,if β∈¯Fk,logαk+12ak(σ2−τ2),if β∈¯Ek.

It is easy to check that the function is continuous and strictly positive.

The next result shows that the limiting log-partition function of the GREM can be computed as the sum of the log-partition functions of the REM’s corresponding to the levels of the GREM.

Theorem 2.1.

For every , the following limit exists in probability and in , for all :

 (2.7) p(β):=limn→∞1nlog|Zn(β)|=d∑k=1pk(β),

where , the contribution of the -th level, is given by (2.6).

Remark 2.2.

Restricting (2.7) and (2.6) to the real temperature case , we obtain, for with ,

Thus, we recovered the previously known formula obtained in [11]; see also [16, 7, 9].

2.2. Heuristics

The reader may find the following heuristics useful. There are three natural guesses on the asymptotic behavior of :

1. expectation dominates: behaves approximately as its expectation; see Figure 3, left. This guess turns out to be correct in phase .

However, it can happen that the fluctuations of around its expectation are of larger order than the expectation. In this case, we end up in the following regime:

1. fluctuations dominate: behaves approximately as its standard deviation; see Figure 3, right. This guess turns out to be correct in phase .

Still, it can happen that due to the presence of heavy tails neither the expectation nor the standard deviation are adequate to estimate the true magnitude of the partition function. In this case, one can make the following guess:

1. extremes dominate: behaves approximately as the maximal summand in (2.1). This guess turns out to be correct in phase .

Summarizing, we arrive at the following three guesses for the limiting log-partition function :

 (2.8) Expectationpk(β) =limn→∞1nlog∣∣EZ(k)n(β)∣∣=logαk+12ak(σ2−τ2), (2.9) Fluctuationspk(β) =limn→∞1nlog√VarZ(k)n(β)=12logαk+akσ2, (2.10) Extremespk(β) =limn→∞1nlogmaxj=1,…,Nn,k∣∣∣eβ√nakη(k)j∣∣∣=|σ|√2aklogαk.

It turns out that these formulae indeed give the correct value of in phases , , , respectively.

2.3. Global limiting distribution of complex zeros

Using Theorem 2.1, it is possible to obtain the limiting distribution of complex zeros of the GREM partition function .

For , the partition function of the REM corresponding to the -th level of the GREM, the limiting distribution of zeros has been computed by Derrida [15]; see also [25] for a rigorous proof. The main idea is to use the Poincaré–Lelong formula (see, e.g., [20, §2.4.1]). It states that the measure counting the complex zeros of any analytic function (which is not everywhere ) can be represented as

 (2.11) {{Zeros}}{f(β):β∈C}=12πΔlog|f(β)|.

Here, is the Laplace operator in the complex -plane. The Laplace operator should be understood in the sense of generalized functions (distributions). Applying this formula to , dividing by , interchanging the large limit and the Laplacian (which should be justified), and using (2.5), one can show that weakly on ,

The distributional Laplacian of (see, e.g., Section 14.3 for the details of the computation), is a measure on given by

 (2.12) Ξk:=Δpk=ΞFk+ΞEFk+ΞEGk,

where , , are measures on the complex plane defined as follows:

1. is times the two-dimensional Lebesgue measure restricted to .

2. is times the one-dimensional length measure on the boundary between and (which consists of two circular arcs).

3. is a measure having the density with respect to the one-dimensional length measure restricted to the boundary between and (which consists of four line segments).

Thus, the zeros of fill the two-dimensional region asymptotically uniformly with density , but some zeros concentrate around the boundary of with one-dimensional density asymptotically proportional to . The term is just the pointwise Laplacian of , whereas the terms and appear because the normal derivative of the function has a jump discontinuity on the boundary of the phase . On the boundary between and , the normal derivative of is continuous, hence this boundary makes no one-dimensional contribution to .

We now proceed to the complex zeros of , the partition function of the GREM. In view of Theorem 2.1, it is not surprising that the limiting distribution of zeros of can be obtained as a superposition of the limiting zeros distributions of the corresponding REM’s.

Theorem 2.3.

The following convergence of random measures holds weakly on the space :

 (2.13) 1n{{Zeros}}{Zn(β):β∈C}w⟶n→∞12πΞ,

where .

2.4. Phase diagram

We can now describe the phase diagram of the GREM in the complex plane; see Figure 4. It is obtained as a superposition of the phase diagrams of the corresponding REM’s. Take some . For every , we can determine the phase (, , or ) to which belongs and write the result in form of a sequence of length over the alphabet . However, it is easy to see that only phases of the following form are possible:

 Gd1Fd2Ed3=G…Gd1F…Fd2E…Ed3,

where are such that . In other words, we have an ordering of the level phases which can be symbolically expressed as

 G≻F≻E.

For example, it is not possible that a level in -phase is followed by a level in - or in -phase. This follows from the fact that if for some , then and for . This ordering of phases agrees with the observation of Saakian [40]. The phases of the GREM are therefore given by

 Gd1Fd2Ed3=(G1∩…∩Gd1)∩(Fd1+1∩…∩Fd1+d2)∩(Ed1+d2+1∩…∩Ed),

where are such that . If , then we say that the levels are in the -phase, the levels are in the -phase, and the levels are in the -phase. Note that each is an open subset of the complex plane. The union of the closures of these sets is the entire complex plane. The total number of phases is . Only of these phases, namely those of the form , intersect the real axis.

2.5. Central limit theorem in the strip |σ|<σ12

In this and subsequent sections, we identify the limiting fluctuations of the partition function . We can view as a sum of random variables in a triangular summation scheme. Although these random variables are dependent (unless ), the limiting distribution of their sum is infinitely divisible, as we shall see. It is well known that an infinitely divisible distribution can be decomposed into a superposition of a Gaussian and a Poissonian component. In this section, we consider the case in which only the Gaussian component is present. The next result states that in the strip the partition function satisfies a central limit theorem.

Theorem 2.4.

Let be such that . Then,

 (2.14) Zn(β)−EZn(β)√VarZn(β)d⟶n→∞{NC(0,1),if τ≠0,NR(0,1),if τ=0.

To draw corollaries from Theorem 2.4, we need to obtain expressions for and . Recall that denotes the variance of , , and . Recall the convention that .

For every , .

Proof.

If is real normal random variable with mean zero and variance , then , . Since every Gaussian random variable in (1.7) has variance , we immediately obtain the required formula. ∎

Next, we establish an asymptotic formula for , as . The asymptotic behavior of the variance displays several regimes (see Figure 5, left) which are separated by the circles

 |β|=σk√2,1≤k≤d.
Proposition 2.6.

Let be arbitrary. For , write

 bk=logα+2σ2a+d∑m=k+1(logαm−|β|2am).

Then,

 VarZn(β)∼⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ebkn,if σk√2<|β|<σk+1√2,1≤k≤d,eb1n,if 0<|β|<σ1√2,2ebkn,if |β|=σk√2,2≤k≤d,eb1n,if |β|=σ1√2.

In the first two cases, the formula holds locally uniformly as long as stays in the specified region.

As an immediate corollary of Proposition 2.6, we obtain the following result comparing the expectation and the standard deviation of .

Proposition 2.7.

For any ,

 limn→∞|EZn(β)|√VarZn(β)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∞, for% |β|<σ1√2,1, for |β|=σ1√2,0, for |β|>σ1√2.

Depending on which quantity, the expectation or the standard deviation, has larger order of magnitude, we can derive from Theorem 2.4 the following two corollaries. The corresponding domains are shown in Figure 5, right.

Proposition 2.8.

If and (which means that with ), then we can drop the expectation in (2.14):

 Zn(β)√VarZn(β)d⟶n→∞NC(0,1).
Proposition 2.9.

If and (which implies but is not equivalent to ), then

 Zn(β)EZn(β)d⟶n→∞1.

If is real, then the result of Proposition 2.9 is contained in [7, Theorem 1.7]. Theorem 2.4 (which is stronger than Proposition 2.9) seems to be new even in the case .

2.6. Central limit theorem for |σ|=σ12

We will show that on the boundary of the strip, i.e. for , the central limit theorem still holds, but with a non-standard limiting variance. In order to have the right “resolution” on the boundary, let us assume that depends on in such a way that for some constant ,

 (2.15) σ(n)=σ12−u2√na1+o(1√n).
Theorem 2.10.

Let be such that is constant and satisfies (2.15). Then,

 (2.16) Zn(β)−EZn(β)√VarZn(β)d⟶n→∞{NC(0,Φ(u)),if τ≠0,NR(0,Φ(u)),if τ=0.

Here, is the standard normal distribution function.

In particular, if does not depend on , then and the variance of the limiting distribution is . For the case of the REM and real , this fact was discovered in [10]. See also [24] for a version with a fine “resolution” as in (2.15). For the case of the REM and complex , see [25]. In the case of the GREM, Theorem 2.16 is new even in the real case. The appearance of the “truncated variance” in (2.16) can be explained as follows. For , the limiting distribution is Gaussian, whereas it turns out that for the first level of the GREM contributes only to the Poissonian component of the limiting distribution. In the boundary case, some energies at the first level of the GREM have left the Gaussian part, but have not arrived yet at the Poissonian part. This is why the variance of the limiting Gaussian distribution is smaller than in the boundary case.

2.7. Poisson cascade zeta function

The fluctuations of in phases of the form with will be described using a random zeta function associated to the Poisson cascades. In this section, we define this function and state results on its meromorphic continuation.

Let be the points of a unit intensity Poisson point process on . The points are always arranged in an increasing order. The Poisson process zeta function is defined by

 ζP(z)=∞∑k=1P−zk,Rez>1.

With probability , the above series converges absolutely and uniformly on compact subsets of the half-plane since a.s. by the law of large numbers. However, with probability , the function admits a meromorphic continuation to the half-plane . Namely, by [25, Theorem 2.6], with probability , we have

 (2.17) ∑Pk≤TP−zk−∫T1t−zdt⟶T→∞ζP(z)−1z−1 on H({Rez>1/2}).

We will need a multivariate generalization of the Poisson process zeta function which will be called the Poisson cascade zeta function. First, we need to define the Poisson cascade point processes; see Figure 6. These and related point processes appeared for example in [7], [39]. Fix dimension . Start with a unit intensity Poisson point process on . Then, for every and every let be a unit intensity Poisson point process on . Assume that all point processes introduced above are independent. Consider the following point process on ,

 (2.18) Π=∑ε=(ε1,…,εd)∈Ndδ(Pε1,Pε1ε2,…,Pε1…εd).

Of course, is not a Poisson process (unless ) since contains infinitely many collinear point with probability . The next lemma states that has the same first order intensity as the homogeneous Poisson process on . It can easily be proven by induction over .

Lemma 2.11.

Let be an integrable or non-negative function on . Then,

 E[∑x∈Πφ(x)]=∫(0,∞)dφ(x)dx.

The random zeta function associated to the Poisson cascade point process is a stochastic process defined by the series

 (2.19) ζP(z1,…,zd)=∑ε∈NdP−z1ε1P−z2ε1ε2…P−zdε1…εd.
Theorem 2.12.

With probability , the series (2.19) converges absolutely and uniformly on any compact subset of the domain

 (2.20) D={(z1,…,zd)∈Cd:Rez1>…>Rezd>1}.

In particular, the function is analytic on with probability .

Theorem 2.12 would be sufficient to treat the GREM at real inverse temperature , as in [7]. However, for complex , we need a meromorphic continuation of to a larger domain.

Theorem 2.13.

With probability , the function defined originally on admits a meromorphic continuation to the domain

 12D={(z1,…,zd)∈Cd:Rez1>…>Rezd>1/2}.

Moreover, the function is analytic on with probability .

We conjecture that with probability there is no meromorphic continuation beyond . In the sequel, we use the notation .

Remark 2.14.

The value of in the case is understood by continuity. In the case , this value is equal to , whereas, for , it is a non-degenerate random variable. (The non-degeneracy follows from the fact that a degenerate random variable cannot satisfy (2.21), see below, with ).

Proposition 2.15.

Consider independent copies of the random analytic function denoted by , . Then, the following distributional equality on holds:

 (2.21)

From Proposition 2.15, we can draw several conclusions about the finite-dimensional distributions of . If , then the distribution of the real-valued random variable is stable with exponent ; see [42, Chapter 1]. In fact, it is even strictly stable meaning that no additive constant is needed in (2.21). If is such that (but are not necessarily real), then the term is real and hence, (which is considered as a random vector with values in ) has a two-dimensional stable distribution (which need not be isotropic); see [42, Chapter 2]. In general, for