STABLE RANDOM FIELDS INDEXED BY FINITELY GENERATED FREE GROUPS 1. INTRODUCTION

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1 STABLE RANDOM FIELDS INDEXED BY FINITELY GENERATED FREE GROUPS SOURAV SARKAR AND PARTHANIL ROY ABSTRACT. In this work, we investigate the extrema behaviour of eft-stationary symmetric α-stabe SαS) random fieds indexed by finitey generated free groups. We begin by studying the rate of growth of a sequence of partia maxima obtained by varying the indexing parameter of the fied over bas of increasing size. This eads to a phase-transition that depends on the ergodic properties of the underying nonsinguar action of the free group but is different from what happens in the case of SαS random fieds indexed by Z d. The presence of this new dichotomy is confirmed by the study of a stabe random fied induced by the canonica action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Suivan. This fied is generated by a conservative action but its maxima sequence grows as fast as the i.i.d. case contrary to what happens in the case of Z d. When the action of the free group is dissipative, we aso estabish that the scaed extrema point process sequence converges weaky to a nove cass of point processes that we have termed as randomy thinned custer Poisson processes. This imit too is very different from that in the case of a attice.. INTRODUCTION A random variabe X is said to foow symmetric α-stabe SαS) distribution α 0,2], the index of stabiity) with scae parameter σ > 0 if it has characteristic function of the form Ee iθx ) = exp{ σ α θ α }, θ R. In this work, we wi aways concentrate on the non-gaussian case, i.e., α 0,2). For encycopedic treatment of α-stabe 0 < α < 2) distributions and processes, we refer the readers to [45]. A random fied X = {X t } t G, indexed by a possiby noncommutative) countabe group G, ) is caed an SαS random fied if for each k, for each t,t 2,...,t k G and for each c,c 2,...,c k R, the inear combination k i= c ix ti foows an SαS distribution. Aso {X t } t G is caed eft-stationary, if {X t } d = {X s t } for a s G. The notion of right-stationarity can be defined anaogousy and wi coincide with eft-stationarity when G is abeian. Whatever we prove for eft-stationary SαS random fieds wi have their corresponding counterparts in the right-stationary case. From now on, we sha write stationary to mean eft-stationary throughout this paper. Thanks to the semina works of Rosiński [34], [35], [36], various probabiistic aspects of stationary SαS random fieds indexed by Z or Z d have been connected to the ergodic theoretic properties of the underying nonsinguar group action; see, for exampe, [37], [26], [43], [33], [44], [6], [42], [4], [5], [28], [3]. For simiar connections in case of max-stabe processes and fieds, we refer the readers to [46], [47], [7], 200 Mathematics Subject Cassification. Primary 60G52, 60G60; Secondary 60G55, 37A40, 20E05. Key words and phrases. Stabe, random fied, extreme vaue theory, point process, nonsinguar group action, free group, boundary action. Sourav Sarkar was supported in part by Loève Feowship at University of Caifornia, Berkeey. Parthani Roy was supported by Cumuative Professiona Deveopment Aowance from Government of India and the project RARE a Marie Curie FP7 IRSES Feowship).

2 [9], [8], [52], [53], [9], [0]. See aso [38], [27], [6], [20] for inks between ergodic theory and stationary infinitey divisibe processes, and [39] for an aternative approach to stabe processes using Maharam systems. In a the works mentioned above, the indexing group G is Z d or R d in the continuous parameter case) for some d and hence amenabe. Many of the proofs use the amenabiity of the underying group in some way or the other. In the present work, we woud ike to go beyond the framework of amenabe groups and study the corresponding stabe random fieds. To this end, we first estabish a genera phase transition resut see Theorem 3. beow) for extremes of stabe fieds indexed by finitey generated countabe groups, and then concentrate on the simpest possibe cass of non-amenabe groups, namey, the finitey generated free groups. We use nonsinguar actions of free groups to construct stationary SαS random fieds in parae to [35, 36] and investigate the extrema properties of such fieds in detais under various ergodic theoretic conditions on the action. The motivation behind our work is twofod. Firsty, ergodic theoretic properties of group actions may change significanty as we pass from amenabe to non-amenabe groups; see, for instance, [50] for a recent artice which shows that the pointwise and maxima ergodic theorems do not hod in L for measurepreserving actions of finitey generated free groups). This necessitates the investigation of the effect of the ergodic theoretic change on various probabiistic aspects of the stabe fieds and finitey generated free groups serve as a convenient test-case in the cass of non-amenabe groups. Keeping this broader goa in mind, we focus on extreme vaue theoretic properties of stationary SαS random fieds indexed by such groups. The second motivation comes from the very simpe observation that by passing to the Cayey graph of the underying free group, we obtain a stationary stabe random fied indexed by a reguar tree of even degree. This, of course, is an important object to study see, for exampe, [29] for a survey on stochastic processes indexed by trees and their importance in probabiity theory, statistica physics, fracta geometry, branching modes, etc.). To our knowedge, the ony famiy of tree-indexed processes with stabe or even heavy taied) marginas was introduced by [, 2] in the form of branching random waks see aso [22], [4], and the more recent works of [3], [25], [23]), [4], [5]. However, the branching random waks are, by design, highy nonstationary. In particuar, no stationary stabe random fied has been constructed on a tree so far and our work can perhaps fi in this gap. An important manifestation of non-amenabiity of free groups is that the usua ba and its interior boundary are asymptoticay proportiona in size. As a resut, compared to the case G = Z d, we indeed observe a different extrema behaviour of {X t } t G when G happens to be a finitey generated free group. In [42, 43], it was shown that a maxima sequence of {X t } t Z d obtained by varying t in d-dimensiona cubes of increasing size) grows faster as we pass from a conservative to a non-conservative Z d -action in its integra representation. In case of finitey generated free groups, we have observed a phase transition behaviour of a simiar maxima sequence and the transition boundary is a different one. In order to confirm the presence of a new dichotomy, we study a cass of stabe random fieds generated by the canonica action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Suivan; see Exampe 3.2 beow. Even though this nonsinguar action is conservative, the maxima of these fieds grow as fast as the maxima in the dissipative case. 2

3 For stationary SαS random fieds generated by dissipative actions of the free group, the corresponding extrema point process has been shown to converge weaky in the space of Radon point measures on [,] \ {0} equipped with the vague topoogy) to a new kind of point process that we have termed randomy thinned custer Poisson process. This imit too is much more sophisticated compared to the corresponding one in the case G = Z d see [33, 4]), where a simpe custer Poisson imit was obtained with no thinning. The presence of thinning in our framework can be expained by the nontrivia contributions of the points coming from the boundary of a ba and hence is ceary a non-amenabe phenomenon. The asymptotic behaviour of the maxima can easiy be read off from the weak convergence of the point process and not surprisingy, the constant term in this imit is much more deicate than the one in the attice case. We woud ike to mention here that the proofs of the main resuts of this paper are not at a straightforward. The proof of Theorem 3., for exampe, reies on the use of ergodic theoretic machineries incuding Maharam extension see [24]) and measurabe union of a hereditary coection see []), and a combinatoria too from geometric group theory. On the other hand, the argument used in proving Theorem 4. is more probabiistic and to some extent anaytic) in nature. Due to the non-amenabiity of free groups, even to estabish that the imiting point process is Radon, we need to give a sharp bound on an expected vaue based on exact counting of vertices see Lemma 6.) that are a specified distance away from the root and in a certain subgraph of the Cayey tree. The paper is organized as foows. Section 2 is devoted to background information on SαS random fieds and their reations to the ergodic theoretic properties of the underying group actions. In Section 3, we present our resuts on the rate of growth of partia maxima for stationary SαS random fieds indexed by genera finitey generated countabe groups, and in particuar by finitey generated free groups. Section 4 deas with the weak convergence of point processes associated with stabe fieds generated by dissipative actions of finitey generated free groups. The resuts in Sections 3 and 4 are proved in Sections 5 and 6, respectivey. The foowing notations are going to be used throughout this paper. For two sequences of positive rea numbers {a n } and {b n }, the notation a n b n wi mean a n /b n as n. On the other hand, for two σ-finite measures m and m 2 defined on the the same measurabe space, m m 2 wi signify that the measures are equivaent. For any σ-finite measure space S,S,m), we define the function space L α S,m) := { f : S R measurabe : f α < }, where f α := S f s) α mds)) /α. For two random variabes X, Y not necessariy defined on the same probabiity space), the notation X d = Y indicates that X and Y are identicay distributed. For two random fieds {X t } t G and {Y t } t G, we write X t fdd = Y t, t G to mean that they have the same finite-dimensiona distributions. 2. BACKGROUND Let G, ) be a countabe group which wi be a finitey generated free group in most cases) with identity eement e and S,S,m) be a σ-finite measure space. A coection of measurabe maps ϕ t : S S indexed by t G is caed a group action of G on the measurabe space S,S) if ) ϕ e is the identity map on S, and 2) ϕ u v = ϕ v ϕ u for a u,v G. 3

4 Note that the order in which the two maps ϕ v and ϕ u appear in the above definition is important because G is mosty going to be a noncommutative free group in this work. A group action {ϕ t } t G of G on S is caed nonsinguar if m ϕ t m for a t G. Here denotes equivaence of measures. Let X = {X t } t G be an SαS 0 < α < 2) random fied indexed by G. Any such random fied has an integra representation of the type 2.) X t fdd = S f t s)mds), t G, where M is an SαS random measure on some standard Bore space S,S) with σ-finite contro measure m, and f t L α m) for a t G. See, for instance, Theorem 3..2 of Samorodnitsky and Taqqu 994) [45]. One can assume, without oss of generaity, that the union t G Support f t ) of the supports of f t is equa to S. If further {X t } t G is stationary, then one can show, foowing an argument of Rosiński see [34], [35], [36]), that there aways exists an integra representation of the foowing specia form ) d m /α ϕt 2.2) f t s) = c t s) d m s) f ϕ t s), t G, where f L α S,m), {ϕ t } t G is a nonsinguar G-action on S, and {c t } t G is a measurabe cocyce for {ϕ t } taking vaues in {,+} i.e., each c t is a measurabe map c t : S {,+} such that for a t,t 2 G, c t t 2 s) = c t2 s)c t ϕ t2 s)) for m-amost a s S). One says that a stationary SαS random fied {X t } t G is generated by a nonsinguar G-action {ϕ t } if it has an integra representation of the form 2.2). A measurabe set W S is caed a wandering set for the action {ϕ t } t G if {ϕ t W) : t G} is a pairwise disjoint coection. It is a we-known resut see, for exampe, [] and [2]) that S = C D, where C and D are disjoint and {ϕ t }-invariant measurabe sets such that ) D = t G ϕ t W ) for some wandering set W, 2) C has no wandering subset of positive measure. This decomposition of S into two invariant parts is known as the Hopf decomposition. D is caed the dissipative part, and C the conservative part of the action, and the corresponding action {ϕ t } is caed conservative if S = C and dissipative if S = D. Another important decomposition is the Neveu decomposition see, for exampe, []) of S into the positive and nu parts of the nonsinguar action as described beow. Foowing Lemma 2.2 and Theorem 2.3 i) in [5] the arguments in the proof appy to a countabe groups, not just Z d ) we decompose S = P N into two {ϕ t }-invariant sets P positive part) and N nu part), where the set P is the argest moduo m) set where one can have a finite measure equivaent to m that is preserved by {ϕ t }, and N is the compement of P. Obviousy P C because a nontrivia wandering set wi never aow a finite invariant measure equivaent to m. A measurabe subset B S is caed weaky wandering if there is a countaby infinite subset {t n : n N} G such that ϕ tn B) are a disjoint. Ceary the positive part P has no weaky wandering set of positive measure. 4

5 Foowing the notations used in [35] and [36], it is easy to obtain the foowing unique in aw decomposition of the random fied {X t } t G as fdd X t = f ts)mds) + f ts)mds) =: X C t + X D t, t G C D into a sum of two independent random fieds X C t and X D t, generated by conservative and dissipative G- actions, respectivey. Note that, foowing the same proof as that of Proposition 3. in [42], if a stationary SαS random fied {X t } t G is generated by a conservative dissipative, resp.) G-action, then in any other integra representation of {X t } the G-action must be conservative dissipative, resp.). Roughy speaking, stabe random fieds generated by conservative actions tend to have onger memory simpy because a conservative action keeps coming back. For G = Z d, this was made precise by studying the rate of growth of partia maxima and imits of sequences of scaed point processes in [43], [33], [42] and [4]. We review their resuts here. Let {X t } t Z d be a stationary SαS random fied and M n := max X t for t n n with being the L -norm. Then as n, n d/α M n Here { 2.3) C α = x sinxdx) α = 0 { C /α α κ X Z α if {ϕ t } t Z d is not conservative, 0 if {ϕ t } t Z d is conservative. α Γ2 α) cosπα/2) if α, 2 π if α =, Z α is a standard Frechét type extreme vaue random variabe with distribution function 2.4) PZ α x) = e x α, x > 0, and κ X is a positive constant depending ony on the random fied {X t } t Z d. In other words, if {X t } t Z d is generated by a conservative action, then the maxima sequence M n grows at a sower rate because onger memory prohibits sudden changes in X t even when t is arge. The foowing resut on weak convergence of a sequence of scaed point processes associated with stationary SαS random fieds on Z d generated by dissipative action is from [33] d = case) and [4] d > case). Assume now that {X t } t Z d is generated by a dissipative Z d -action. In this case, we can assume without oss of generaity that {X t } t Z d has the foowing mixed moving average representation in the sense of [48]): fdd 2.5) X t = f w,s t)mdw,ds), t Z d, W Z d where f L α W Z d,ν ζ), ζ is the counting measure on Z d, ν is some σ-finite measure on the standard Bore space W,W ), and M is a SαS random measure on W Z d with contro measure ν ζ; see [36] and [42] for detais. Suppose ν α is the symmetric measure on [,] \ {0} such that ν α x,] = ν α [, x) = x α for a x > 0. Let 2.6) δ ji,w i,u i ) PRMν α ν ζ) i 5

6 be a Poisson random measure on [,] \ {0} ) W Z d with mean measure ν α ν ζ. Then {X t } t Z d in 2.5) has the foowing series representation ignoring a factor of C /α α ): 2.7) fdd X t = j i f w i,u i t), t Z d, i It was shown in [33] and [4] that in the space M of Radon measures on [,]\{0} endowed with vague topoogy), δ 2n) d/α X t Ñ, t n which is a custer Poisson random measure with representation 2.8) Ñ = δ ji f w i,t) ui =0), i= t Z d where j i,w i,u i are as in 2.6). The Lapace functiona of the above Ñ is E e Ñ g) ) { { }) } = exp exp gx f w,t))) ν α dx)νdw), W x >0 t Z d for a measurabe g : [,] \ {0} [0,). Here Ñ g) denotes the random variabe obtained by integrating g with respect to the random measure Ñ. Note that in the representation of the custer Poisson random measure N given in Theorem 3. in [4], the term ui =0) was missing even though the computation of the imiting Lapace functiona was correct. A simiar comment appies to Theorem 3. of [33]. Reca that for any finitey generated countabe group G with a symmetric w.r.t. taking inverses) generating set D not containing the identity eement e, the Cayey graph V,E) consists of the vertex set V = G and edge set E = {u,v) : u v D}. Ceary, symmetry of D turns this into an undirected graph and e / D impies there is no sef-oop. In this paper, we sha use the anguage of Cayey graphs to investigate the asymptotic behaviours of a sequence of partia maxima and a sequence of point processes associated with the stationary SαS random fieds indexed by finitey generated free groups. In most of the discussions beow, G wi denote a free group of finite rank d 2 except in Theorem 3., where G wi simpy be a genera finitey generated countabe group) with the generating set D = {a,a,a 2,a 2,...,a d,a d } being the coection of d independent symbos and their inverses. This group consists of a reduced words formed out of the symbos in D with the operation being concatenation foowed by reduction and its Cayey graph is a 2d-reguar tree. See, for exampe, [2] for detais on free groups and Cayey graphs. For any t G, we define t to be the graph distance of t from the root e in the Cayey graph of the group G, i.e., t = dv,e), where da,b) denotes the graph distance between vertices a and b in the Cayey graph of G. Aso 2.9) E n := {t G : t n}, and C n := {t G : t = n} denote the ba of radius n and its interior boundary, respectivey. When G is a free group of finite rank d 2, an easy counting yieds that E n = + d d [2d )n ] = Θ2d ) n ) and C n = 2d)2d ) n for a n. In particuar, C n is asymptoticay proportiona to E n, which is a manifestation of nonamenabiity. As a resut, the extreme vaues of stabe random fieds indexed by finitey generated free 6

7 groups are affected by the significant contributions from the interior boundary of E n. This wi become cear in Sections 3 and 4 beow. In the next section, we sha study the asymptotic behaviour of the partia maxima sequence of 2.0) M n = max t E n X t, n of the stationary SαS random fied {X t } t G obtained by restricting the fied to the ba E n. As we sha see, there wi be a phase transition as ong as G is a finitey generated countabe group. Of course, for G = Z d, the phase transition boundary has to coincide with the Hopf boundary. However, when G is a free group of finite rank d 2, non-amenabiity of the group wi induce a new transition boundary that ies stricty between the Hopf and Neveu boundaries. 3. RATE OF GROWTH OF PARTIAL MAXIMA Let G be a countabe group generated by a finite symmetric set D and {X t } t G be a stationary SαS random fied having an integra representation of the form 2.), where f t is given by 2.2). We sha eventuay speciaize to the case when G is a free group of finite rank d 2 and investigate the extreme vaue theory of the fied. Define E n and C n as in 2.9) and the partia maxima sequence M n by 2.0). We define a deterministic sequence 3.) b n = b n f ) = max t E n f t x) α mdx) ) /α, n =,2,..., where f L α S,m) is used in the definition of f t in 2.2). Note that by Coroary of [45], for any specific random fied {X t } t G, the quantity b n does not depend on the choice of f t in its integra representation 2.). However, in this artice, we sha anayze a cass of stationary SαS random fieds obtained by varying f L α S,m), and fixing the group action {ϕ t } t G and the cocyce {c t } t G in 2.2). With this viewpoint in mind, we are introducing the notation b n f ) even though in many situations, we sha stick to b n. Foowing arguments simiar to that in [43], one can show that to a arge extent, the asymptotic behaviour of the random sequence M n is determined by that of b n. Hence we first ook at the growth rate of the deterministic sequence b n and use that to anayze the same for M n. In the seque, c wi aways denote a positive constant that may not necessariy be the same in each occurrence. Theorem 3.. Let G be a countabe group generated by a finite symmetric set, and {ϕ t } t G be a nonsinguar group action on a σ-finite standard measure space S,S,m). i) Then the set S can be uniquey decomposed into two disjoint {ϕ t }-invariant measurabe sets A and B i.e. S = A B) such that, for any f L α S,m), b a) whenever f is supported on B, im n f ) n = 0, and E n /α b b) if the support of f has some nontrivia intersection with A, then imsup n f ) n > 0. E n /α The above decomposition is the same for a measures equivaent to m. ii) The dissipative part D A, and the positive part P B. iii) If the SαS random fied is given by the integra representation 2.) and 2.2), then X t fdd = f ts)mds) + A f ts)mds) =: X A t B 7 + X B t, t G

8 can be written as a sum of independent random fieds Xt A and Xt B such that the foowing resuts hod. a) If the component Xt A is zero, then M n / E n /α P 0 b) If the component Xt A is nonzero, then M n = O p E n /α ) i.e., M n / E n /α is tight), and there exists a subsequence M nk of M n and a positive constant c > 0 such that M nk / E nk /α cz α, where Z α is an α-fréchet random variabe with distribution function given in 2.4). Keeping in mind the first part of the above theorem, we sha ca A the nondegenerate part and B the degenerate part. It is possibe that this decomposition may be known in the ergodic theory iterature by some other name athough our extensive iterature search did not revea any. When G = Z d, the above decomposition is the same as the Hopf decomposition of the group action with A = D and B = C see [43] and [42]). For a genera group G, even if the support of f has a nontrivia intersection with the nondegenerate part A, one cannot surey say that im n b n f ) E n /α > 0 simpy because the imit may not aways exist. In particuar, we need to work with imit superior as opposed to the imit in Part i)b) of Theorem 3.. However, when G is a free group of finite rank d 2, we can significanty improve our previous resut as shown in the foowing theorem. Theorem 3.2. When G is a free group of finite rank d 2, for any f L α S,m) whose support has some nontrivia intersection with A, one has, im inf n b n f ) > 0. E n /α Aso given any subsequence M nk of M n, there exists a further subsequence M nk and a positive constant c > 0 such that M nk / E nk /α cz α, where Z α is an α-fréchet random variabe as before. In fact, if f is supported on the dissipative part D, then for any finitey generated countabe group G, iminf n b n f ) E n /α > 0 see the proof of Part ii) of Theorem 3.). When G is a free group of finite rank d 2 and Support f ) D, then the imit exists and as a consequence, M n /2d ) n/α cz α for some c > 0; see Coroary 4.2 beow. For the rest of this section and the next one, we sha assume that G is a free group of finite rank d 2. Thanks to the non-amenabiity of this group, the decomposition of S into degenerate and nondegenerate parts is now different from what happens in the Z d case, where it coincides with the Hopf decomposition. This eads to a new dichotomy see beow) for the maxima sequence M n defined in 2.0). Theorem 3.3. When G is a free group of finite rank d 2, there exists a stationary SαS random fied indexed by G generated by a conservative action, for which we have M n /2d ) n/α C /α α Z α, where C α is as defined in 2.3) and Z α is a standard α-fréchet random variabe. Moreover if CN := C N denotes the conservative nu part of the action, then C N can have nontrivia intersections with both the nondegenerate part A and the degenerate part B. 8

9 That is, we sha give two instances see Exampes 3.2 and 3.3 beow) of stationary SαS random fieds generated by conservative nu actions, such that for one, the partia maxima grows at the rate of 2d ) n/α or E n /α ) and for the other, the partia maxima grows at a stricty smaer rate. Note that Hopf and Neveu Figure. Boundary between nondegenerate part A) and degenerate part B) decompositions of the underying nonsinguar action induce the partition of S = P CN D into positive, conservative nu, and dissipative parts. Our phase transition boundary between the degenerate and the nondegenerate parts) ies stricty between the Hopf and Neveu boundaries and passes through the conservative nu part CN ) of the group action; see the dotted ine in Figure. The next resut says that the asymptotic behaviour of the partia maxima for the bas of increasing radii is actuay determined by the interior boundaries of the bas. Ceary, this is intrinsicay a non-amenabe phenomenon that woud never happen in the attice case. Theorem 3.4. Let G be a free group of finite rank d 2, and et the stationary SαS random fied indexed by G has integra representation 2.). Then we have maxt En f t x) α mdx) maxt Cn f t x) α mdx) im sup n 2d ) n > 0 if and ony if imsup n 2d ) n > 0. In the next theorem, we try to find some sets that beong to the nondegenerate part A of Theorem 3.. It states that if a set has sufficient number of disjoint transates in each ba, then the set is inside A. Theorem 3.5. Define, for any subset B S, a n B) to be the maximum number of sets in {ϕ t B) : t E n } that are pairwise disjoint, i.e., a n B) := max{ T : T E n and ϕ t B) are pairwise disjoint for a t T }. If imsup n a n B) E n > 0 for some subset B S, then B A. The proofs of the theorems stated in this section are given in Section 5. Finay, we give three exampes of stationary SαS random fieds generated by conservative actions. The first exampe hods for any countabe finitey generated group G, and is crucia for the proof of Part iii) of Theorem 3.. This is parae to Exampe 5.4 in [43]. Exampe 3.. Let S = R G and M is an SαS random measure on R G whose contro measure m is a probabiity measure under which the projections π t,t G) are i.i.d. random variabes with a finite absoute α th moment. Let π = π e : R G R as πx t ) t G ) = x e, and ϕ t is the shift operator, i.e., ϕ t x s ) s G )) k = x t k. 9

10 Ceary this action is probabiity m-preserving and hence is conservative. The random fied has the integra representation fdd X t = π ϕ t dm = π t dm, t G. R G R G Now, if the projections π t, t G are i.i.d. Pareto random variabes with mπ e > x) = x θ, x for some θ > α, then as in Exampe 5.4 in [43], we get, b n c /α α,θ E n /θ as n for some positive constant c α,θ. Furthermore, in this case, M n / E n /θ converges to an α-fréchet distribution. This exampe wi be required in the proof of Part iii) of Theorem 3.. In the next two exampes, G is a free group of finite rank d 2. The first one considers the canonica action of the free group on its Furstenberg-Poisson boundary with the Patterson-Suivan measure on it. Any stationary SαS random fied generated by this nonsinguar action satisfies iminf n b n / E n /α > 0 even though the action is conservative. Exampe 3.2. The boundary G of the group G consists of a infinite ength reduced words made of powers of symbos from the generating set D. Given a group eement g G\{e}, define H g G) to be the cyinder set consisting of a infinite words starting with g, i.e., H g = {ω G : [ω] g = g}, where [ω] n represents the eement in G formed by the first n-ength segment of ω. Define S to be the σ- fied on S = G generated by the cyinder sets H g, g G \ {e}. It is easy to check that there exists unique probabiity measure m on S,S) such that mh g ) = 2d2d ) g for a g G \ e. This measure is known as the Patterson Suivan measure see [30]) and it turns S = G into a Furstenberg- Poisson boundary see [49]) of the group G. The free group G acts canonicay on S,S,m) in a nonsinguar fashion by 3.2) ϕ t ω) = t ω, for t G, ω S, where is the eft-concatenation of a finite word with an infinite word foowed by reduction. The Radon- Nikodym derivatives of this action are given by dm ϕ t dm ω) = 2d ) B ωt), t G, ω S, where B ω t) = t 2 t ω the Busemann function associated with ω) with t ω being the ongest common initia segment aso known as the confuent) of t and ω. For further detais on the boundary action, we refer the reader to [5], where it was estabished that this action is conservative. We sha first show that the boundary action is nu, i.e., its positive part in the Neveu decomposition) is empty moduo m. To this end, et D = {a,a,a 2,a 2,...,a d,a d } be the generating set of G with d independent symbos and their inverses as before. Then take the cyinder set H a and consider the sets 0

11 ϕ t H a ) for t = e and t = a xa k, where k = 0,,2,... and x D \ {a,a }. It is easy to see that, for is a weaky wandering set, and a such t, ϕ t H a ) are disjoint and their union is G. This shows that H a hence the boundary action is nu. Define f t by 2.2) with the constant function f on S, the trivia cocyce c t for a t G, and the boundary action 3.2). Then {X t } t G defined by the integra representation 2.) is a stationary SαS random fied generated by a conservative nu action. We now caim that 3.3) F n ω) := max f t ω) α dm ϕ t = max t E n t E n dm ω) = max2d ) Bωt) = 2d ) n t E n for a ω S. We need to show the ast equaity above. As t E n, for any ω S, the ength of the confuent t ω t n. Hence n B ω t) n for a t E n, ω S. Hence F n ω) 2d ) n. To see the other inequaity, note that, for any fixed ω S = G, if we take g = [ω] n C n E n, then g ω = n, so that B ω g) = g 2 g ω = n 2n = n. Hence, 2d ) Bωg) = 2d ) n, so that F n ω) 2d ) n. This proves 3.3). Hence, b n f )) α = F n ω)mdω) = 2d ) n, and, S b n f )) α b n f )) α d im = im = > 0. n E n n d d 2d )n d Foowing the arguments in the proof of Theorem 4. in [43], one has M n C/α 2d ) n/α α Z α, where Z α is a standard α-fréchet random variabe defined in 2.4). Our next caim is that the degenerate part B of the boundary action is an m-nu set. To estabish this, take the cyinder set H a for any a in the generating set D, and the set T n = {a g : g C n }. Ceary T n E n as a g a + g = n, and {ϕ t H a )} t Tn = {H g a a : g C n } = {H g : g C n } are a the cyinder sets of dimension n which are a disjoint. Since T n = C n 2d 2 E 2d ) 2 n as n, by Theorem 3.5, we have, H a A. As this happens for a symbos a D, and the union of the cyinder sets over a a D is S, one gets that the nondegerate part A = S moduo m. Remark 3.6. Note that the boundary action defined here differs sighty from that defined in [5], where the authors define ϕ t ω) = t ω, for t G, ω S. We use the definition in 3.2) so as to match with our convention for group actions used in this paper, i.e, ϕ u v = ϕ v ϕ u for a u,v G. This adjustment does change the Radon-Nikodym derivatives but does not compromise the nonsinguatrity or the conservativity) of the action.

12 The above exampe shows that there exist stationary SαS random fieds generated by conservative nu actions, for which the maxima grows at the rate of 2d ) n/α. But this is not necessariy the case for a such actions. The next exampe shows that there exists a stationary SαS random fied generated by a nu conservative action, for which the maxima sequence grows at a stricty smaer rate. Exampe 3.3. Let the free group G of rank d 2 be generated by the set D as in Exampe 3.2. Take S = R with m = Lebesgue measure, and the group action {ϕ t } t G to be the one that makes a shift of by the action of a and is fixed by the actions of a 2,a 3,...a d. In other words, for a i =,2,...,d, ϕ ai x) = x + {i=}, x R. This group action is ceary measure preserving. Therefore b n 0,] )) α = Leb n,n + ]) = 2n +, and hence b n 0,] )) α / E n 0 as n. Again, the set 0,] is weaky wandering, as ϕ t 0,]) for t = a k, k Z are a disjoint, and their union is the whoe set R. As the set 0,] B the degenerate part - reca Theorem 3.) and B is ϕ t -invariant, it contains a transates {ϕ t 0,]),t = a k }, and hence B = R. Hence this action is conservative, nu and yet degenerate. By Theorem 3.3, for any stationary SαS random fied generated by this action, the partia maxima satisfies M n /2d ) n/α P DISSIPATIVE CASE: POINT PROCESS AND MAXIMA We woud ike to begin this section by observing that the representations 2.5) and 2.7) can be generaized to any countabe group G, not just Z d. More specificay, one can estabish that for any countabe group G, a stationary SαS random fied {X t } t G is generated by a dissipative G-action if and ony if it has a mixed moving average representation of the form 4.) X t fdd = W G f w,t s)dmw,s), t G, where M is an SαS random measure on W G with contro measure ν ζ, and ν is a σ-finite measure on the measurabe space W,W ), ζ is the counting measure on the group G, and f L α W G,ν ζ) as mentioned in Section 2, this terminoogy was introduced in [48]). See [40], where the argument is given for any countabe abeian group extending the works of [36] and [42]. With a itte bit of care about the side of mutipication, etc.), such an argument can be carried forward to any countabe group, not necessariy abeian. As in Section 2, taking ν α as the symmetric measure on [,] \ {0} satisfying ν α x,] = ν α [, x) = x α for a x > 0, 4.2) N = δ ji,v i,u i ) PRMv α ν ζ) i on [,] \ {0} ) W G, and dropping a factor of C /α α, one can obtain the series representation fdd 4.3) X t = j i f v i,t u i ), t G. i In this section, we sha assume that G is a free group of finite rank d 2 and study the weak imit of scaed point process and partia maxima sequences induced by a stationary SαS random fied 4.3) generated by a dissipative and hence nondegenerate by Part ii) of Theorem 3. above) action. Thanks to the nontrivia 2

13 contributions see, for instance, Theorem 3.4 above) coming from the interior boundary C n of E n as a resut of the non-amenabiity of G, these imits are different from those arising in the case of Z d. The cass of point process imits that we obtain are competey nove and we have termed this new cass as randomy thinned custer Poisson processes. We woud ike to mention once more that in case of Z d, Poisson custer processes arise as imits and the random thinning phenomenon is absent; see [33], [4]. We wi state our resuts for the random fied 4.3) after defining various quantities that appear in the statement of the main theorem of this section. 4.. Construction of -subgraphs. For each fixed Z, we define a cass of subgraphs of the Cayey graph of G by specifying the set of vertices of each subgraph. We ca them -subgraphs, and denote the set of a -subgraphs by Γ. We sha consider three cases and in each case, we sha construct a typica -subgraph as described beow. Reca that for u,v G, du,v) denotes the graph distance between the vertices u and v in the Cayey graph of G, v = dv,e), and C n denotes the interior boundary of the ba E n of size n. Case : = 0. Consider a sef-avoiding path starting from the root e. Let the vertices aong the path be v 0 = e,v,v 2,..., where v k = k. For each such vertex v k, we define a coection of sets of vertices V k by 4.4) V k = {t G : dt,v k ) k}, k = 0,,2,.... Note that {e} = V 0 V V A typica -subgraph for = 0) corresponding to a particuar sefavoiding path {v 0 = e,v,v 2,...} is defined as the union of a these sets of vertices i=0 V i. The coection of a such subgraphs corresponding to a sef avoiding paths starting from the root e is the set Γ 0. Case 2: > 0. Here we consider a sef avoiding paths {v 0,v,v 2,...} starting from some vertex v 0 C that goes away from the root, i.e., v 0 =, v = +, v 2 = +2 and so on. For any such sef avoiding path, define the coection of vertices V k by 4.4), and we have the corresponding typica -subgraph as i=0 V i. The coection of a such subgraphs is denoted by Γ. Case 3: < 0. For any fixed < 0, consider a sef avoiding paths {v 0,v,v 2,...} starting from some vertex v 0 C such that v 0 =, v =, v 2 = 2,...,v = e, v + =, v +2 = 2 and so on. Given such a path, we define, once again, the corresponding -subgraph to be i=0 V i, where V k is as in 4.4). The coection of a such subgraphs corresponding to a sef avoiding paths is our Γ. Given g L α W G,ν ζ), Z and ξ Γ, we define functions g,ξ) by appropriatey thinning the function g to the -subgraph ξ Γ, i.e., 4.5) g,ξ) w,t) = gw,t) {t ξ}, w W, t G An a-encompassing Poisson random measure. Next we sha describe for each Z, a probabiity measure γ on the set Γ of a -subgraphs as a uniform measure on a -subgraphs. We sha construct these by resorting to Komogorov consistency theorem. To this end, first fix Z. For any m N, we say that two -subgraphs are m-essentiay distinct if the two subgraphs when restricted to E m are distinct. We denote, by Γ m), the finite set of a m-essentiay distinct -subgraphs. Define X = {,2,...,2d}. We caim that for each,m) Z N, the set Γ m) can be embedded into C X m. To see this, note that any two essentiay distinct subgraphs in Γ m) wi necessariy correspond to 3

14 two distinct sef avoiding) paths of ength m starting from some vertex in C. But the path associated to such a subgraph may not be unique, in that case, we just choose any one of the associated paths. However, for any two distinct subgraphs, any two corresponding paths associated to them wi necessariy be distinct.) And since the degree of each vertex in G is 2d, any such path is an eement of C X m. Simiary, Γ can be embedded into C X. Once again, fix Z. Now suppose that γ m) is the uniform distribution on Γ m) embedded in C X m ). Then ceary {γ m) } m is a consistent system of probabiity measures. Therefore by Komogorov consistency theorem, we get a unique probabiity measure γ on Γ embedded in C X ), such that γ restricted to Γ m) is γ m) for each m N. Now that we have defined the sets of -subgraphs Γ and the measures γ on them, we consider the product probabiity space Γ =, γ = ZΓ γ ). Z And as each Γ is embedded in a compact separabe metric space, so is their product Γ. In particuar, Γ is ocay compact and separabe. We define a sequence of i.i.d. Γ-vaued random variabes r i = r i, : Z ), i N with common aw γ and independent of the Poisson point process N defined in 4.2). We aso take a coection of i.i.d. integervaued random variabes s i, i N independent of N and {r i } i N, and distributed according to the probabiity measure µ on Z defined by 4.6) µ{k}) = By Proposition 3.8 of [3], { 2d2d ) k ) d d if k = 0,, 2,..., 0 otherwise. 4.7) M = δ ji,v i,u i,s i,r i ) PRMv α ν ζ µ γ) i on [,] \ {0} ) W G Z Γ The weak convergence resuts. Let M be the space of a Radon measures on [,]\{0} equipped with the vague topoogy. Since E n = Θ2d ) n ), one expects 2d ) n/α to be the correct scaing in this case. As we sha see, the partia maxima sequence 2.0) grows in this rate as we. Define the function f L α W G,ν ζ) based on f, as f v,t) = f v,t ) for a v W,t G. Using 4.5), define for each Z and for each ξ Γ, the function f,ξ) on W G by f,ξ) w,t) = f w,t) {t ξ} = f w,t ) {t ξ}. With these notations and machineries, we can now state the main theorem of this section. See Section 6 for the proofs of a the resuts stated in this section. 4

15 Theorem 4.. Let {X t } t G be the mixed moving average given in 4.3), and define the sequence of point processes 4.8) N n := t E n δ 2d ) n/α X t, n =,2,.... Then N n N as n ) weaky in the space M, where N is a randomy thinned custer Poisson random measure with representation 4.9) N = i= k G δ j i f u i,r i, ui ) vi,k) ui e) + i= k G δ d d ) /α j i f s i,r i,si ) vi,k) u i =e). Here j i,v i,u i,s i,r i are as in 4.7). Furthermore N is Radon on [,] \ {0} with Lapace functiona 4.0) E e N g) ) { = exp ) ) } 2d 2d ) e k G gx f,ξ) v,k)) γ dξ) ν α dx)νdv) for any nonnegative measurabe function g defined on [,] \ {0}. Z As mentioned earier, in case of G = Z d, the thinning of the function f is absent due to amenabiity of the group. Note that in the above imit, index,ξ) Z Γ of the thinned function f becomes random. That is why we have come up with the term randomy thinned custer Poissson process for the imiting point process N. We can use the convergence of the point process to get the weak convergence of partia maxima M n scaed by2d ) n/α. The imit is a positive constant times the standard α-fréchet distribution and the constant is, not surprisingy, much more sophisticated and invoved compared to the corresponding one in case of Z d obtained in [43] and [42]. Coroary 4.2. Let M n be as in 2.0). Then 2d ) n/α M n C /α α K X Z α, where Z α is a standard α-fréchet random variabe, C α is the stabe tai constant given in 2.3), and α /α K X = 2d)2d ) 2 sup f v,k) ),ξ) γ dξ)νdv)) 0,). Z W Γ k G 4.4. A specia case with eve symmetry. The above theorem takes a particuary simpe form if we assume a eve symmetry assumption on the function f, i.e., if for each v W,t G, 4.) f v,t) = qv, t ), for some function q on W N. For each Z, fix ξ Γ. Observe that by eve symmetry, the thinned functions f,ξ ) and f,ξ ) are equa. We abuse the notation sighty and denote both of these functions by f ). To carify the intriguing but utimatey compex structure of the imiting point process obtained in this section, we present pictures of the -subgraphs see Figure 2) corresponding to f ) for = 0,, when G = Z Z is a free group of rank d = 2 and f satisfies the eve symmetry assumption 4.). These pictures and the coroary beow iustrate what random thinning means in this specia case. 5

16 Figure 2. The -subgraphs corresponding to f ) for = 0,, Coroary 4.3. Let {X t } t G be the mixed moving average given in 4.3), where f satisfies the assumption given in 4.). Define the sequence of point processes N n by 4.8). Then 4.2) N n N := i= t G δ ji f u i ) v i,t) u i e) + i= t G δ d d ) /α j i f s i ) v i,t) u i =e), where j i,v i,u i are as in 4.2), and {s i } are distributed independent of j i,v i,u i ) according to the probabiity measure µ as defined in 4.6). N is Radon on [,] \ {0} with Lapace functiona 4.3) E e N g) ) { = exp ) } 2d2d ) e t G gx f ) v,t)) ν α dx)νdv). = Note that if we assume f satisfies the eve symmetry assumtion 4.), then it is easy to check that the Lapace functiona given in 4.0) reduces to the one in 4.3). As observed earier, f = f, and the expression k G gx f,ξ) v,k)) in the exponent of the Lapace functiona in 4.0) is the same for a ξ Γ. So the inner integra in 4.0) does not depend on the subgraph ξ. Since γ is a probabiity measure, the rest foows. Using f L α W G,ν ζ) and 4.), for ν-amost a v W, we define functions h v L α G,ζ) as foows. If sup t G f v,t) is attained at C k for some k, then assign h v C = f {v} Ck for a 0 k. Next, if sup t / Ek f v,t) is attained at C k for some k > k, then define h v C = f {v} Ck for a k + ) k, and so on. The constant K X in Coroary 4.2 takes the foowing simpe form under the assumption 4.): K α X = 2α d W Lv) α νdv) + W 2h v α α νdv), where Lv) := sup t G f v,t) and for any function g L α G,ζ), g α = t G gt) α ) /α. Note that the first term of K α X was present up to a constant mutipe) in case of Zd see [43] and [42]) but the second term is new and can be interpreted as the contribution of non-amenabiity of the group G) to the custering of the extremes of {X t } t G Open probems. We woud ike to mention that the resuts in this paper give rise to a bunch of open probems some of which wi perhaps be taken up as future directions by the authors. For instance, Gennady Samorodnitsky asked the foowing question in a persona communication with the second author: is it possibe to characterize a finitey generated countabe groups for which the degenerate-nondegenerate decomposition is different from the Hopf decomposition? Whie we beieve that this is perhaps a difficut question, it does open a Pandora s box fu of open and interesting probems. For exampe, it may sti 6

17 be possibe to partiay answer this question by considering specia cases and eventuay giving various sufficient conditions on the group so that a new transition boundary is obtained in Theorem 3.. Most of the works mentioned in the second paragraph of Section have not been extended to the case of random fieds generated by free groups. These can aso ead to many intriguing open probems reating ergodic theory of nonsinguar actions of free groups) with probabiity theory of tree-indexed random fieds). The non-amenabiity of free groups woud surey affect various stochastic properties of such fieds as we and it woud be fascinating to anayze them. In particuar, construction and investigations of max-stabe random fieds indexed by trees wi surey turn out to be important in spatia extremes. Since nonsinguar aso caed quasi-invariant) actions arise naturay in the study of Lie groups, one can think of going beyond countabe groups and R d ), and ask simiar questions for stationary stabe and maxstabe random fieds indexed by Lie groups. Using the structure theorem of abeian groups, [42] gave finer asymptotics for the partia maxima of stabe random fieds indexed by Z d see aso [7] for the continuous parameter case). However such finer resuts are sti missing in our setup mainy due to unavaiabiity of a genera structure theorem for finitey generated noncommutative groups. It is perhaps possibe to resove this issue in specia casses of groups. 5. PROOFS OF THE RESULTS STATED IN SECTION 3 Proof of Theorem 3.. Let us define for any f L α S,m), ψ f ) := imsup n b n f )) α, E n where b n f ) is as defined in 3.). For any measurabe B S with mb) <, et ψb) := ψ B ). Note that for any A,B S with ma B) <, 5.) ψa B) ψa) + ψb), and if 0 f g L α S,m) then 5.2) ψ f ) ψg). We wi need the foowing emmas. Lemma 5.. If f,g L α are such that f x) gx) α mdx) < ε, then ψ f ) ε + ψg) for α 0,) ψ /α f ) ε /α + ψ /α g) for α [,2) Proof. Note that b n f g)) α t E n f t g t )x) α mdx) = E n f x) gx) α mdx) < E n ε, 7

18 from which this resut foows using the triange inequaity, and the facts that x + y) α x α + y α for a α 0,) and x,y 0, and that for α [,2), L α is a normed space. The above emma has the foowing important consequences. Coroary 5.2. If B S with mb) < can be decomposed as B = n= B n, where B i s are pairwise disjoint satisfying ψb n ) = 0 for a n =,2,..., then ψb) = 0. Proof. As mb) = n= mb n) <, hence, given any ε > 0, we can get a sufficienty arge N N such that mb \ N n= B n) < ε. Appying Lemma 5., for α 0,], we get ψb) ε + ψ N n=b n ) ε + N n= ψb n ) by 5.). As ψb n ) = 0 for a n, ψb) ε. Since this hods for a ε > 0, we are done. Coroary 5.3. If ψb) > 0 for a subsets B with 0 < mb) <, then ψ f ) > 0 for a nonzero f L α. Aso if ψb) = 0 for a subsets B with mb) <, then ψ f ) = 0 for a f L α. Proof. For any nonzero f L α, there exists c > 0 and some set C with 0 < mc) < such that f c C. Hence if ψb) > 0 for a B with 0 < mb) <, then ψc) > 0. Thus, using 5.2), ψ f ) c α ψc) > 0. Next assume ψb) = 0 for a subsets B with mb) < and f L α. Then given any ε > 0, we get K arge enough, and c sma enough, such that f x) f x) {c f K} x) α mdx) < ε. Then appy Lemma 5. and note that ψ f {c f K} ) K α ψ{c f K}) = 0 as mc f K) m f c) <. This emma tes us that it is enough to compute ψb) for a sets B with mb) < instead of a functions in L α. The next emma reates ψa) with ψϕ t A)). Lemma 5.4. If ψa) = 0 for some subset A with 0 < ma) < and g G is such that mϕ g A)) <, then ψϕ g A)) = 0. Proof. First assume {ϕ t } is measure m preserving. Then ) ) b n ϕ g A ))) α = m ϕ t ϕ g A)) = m ϕ g.t A) m ϕ t A). t E n t E n t E n+ g The ast inequaity foows as g t g + t n + t and hence {g t t E n } E n+ g. Thus ) b n ϕ g A ))) m α t E ϕ n+ g ta) ψϕ g A)) = imsup im sup n E n n E n ) m t E ϕ n+ g ta) im sup E n E n+ g g = E g ψa) = 0. 8

19 Here we have used the foowing combinatoria fact from geometric group theory: for any finitey generated group G, E m+n E m E n for a m,n N; see Chapter 6 of [8]. Now we assume ϕ t is any nonsinguar map not necessariy measure preserving). We have 0 < ma) <, mϕ g A)) < and ψa) = 0. Define 5.3) w t s) := d m ϕ t s), t G,s S, d m and the group action ϕt of G on S 0,),m Leb) as ) ϕt y s,y) := ϕ t s),, s S, y > 0, t G. w t s) It is easy to see that ϕt preserves the measure m Leb this action is caed Maharram extension; see [24] and Chapter 3.4 of []). Denote for any set B S 0,), ψ B) as before but using the group action ϕt. Aso note that, for any n {0,,2,...}, and any subset B S, ψb) = ψ B n,n + ]). This is because ) m Leb ϕt B n,n + ]) = maxϕ t s) B,nw t s) < y n + )w t s))dymds) t E S 0 t E n n = maxw t s) B ϕ t s))mds) = b n B )) α. t E n S Hence ψ A n,n + ]) = ψa) = 0 for a n = 0,,2,... Aso as ψ ϕ g A) 0,]) = ψϕ g A)), so we need to prove ψ ϕ g A) 0,]) = 0. To this end, et us define Ω n = ϕ ga n,n+]) and B = ϕ g A) 0,]. Then n=0 Ω n is equa to ) ϕ ga n,n + ]) = ϕ g A n,n + ]) = ϕ ga 0,)) = ϕ g A) 0,). n=0 n=0 As B n=0 Ω n, Ω n s are disjoint, so B can be decomposed as B = n=0 B Ω n). Aso m LebB) = mϕ g A)) <, hence by Coroary 5.2 and 5.2), it is enough to show ψ Ω n ) = 0 for a n. Now ψ Ω n ) = ψ ϕ ga n,n + ])) = 0 using the aready considered case of measure preserving actions and ψ A n,n + ]) = ψa) = 0. We are now in a position to present the proof of Theorem 3.. i) For simpicity, assume without oss of generaity, that the contro measure m is a probabiity measure. This can aways be done because if ν is a probabiity measure equivaent to m, define h = f dm dν )/α L α S,ν), and write X t as an integra representation in 2.) and 2.2) repacing f by h and the SαS random measure M by an SαS random measure with contro measure ν. Note that the supports of f and h are equa. Aso b n f ) cacuated with respect to the measure m is same as b n h) corresponding to the measure ν. Henceforth we assume m is a probabiity measure. Consider a subsets B S such that ψb) = 0. As these sets form a hereditary coection i.e., C B and ψb) = 0 impies ψc) = 0), we can take the measurabe union of a such sets, and ca it B. Define A := S \ B. Note that ψb) = 0 by Coroary 5.2 and exhaustion emma see page 7 of Aaronson []). Hence, for any C S, ψc) = 0 if and ony if C B. Consequenty, C A if and ony if for a subsets B C with mb) > 0, one has ψb) > 0. 9