aa r X i v : . [ m a t h . P R ] J a n LARGE DEVIATION FOR OUTLYING COORDINATESIN β ENSEMBLES
THOMAS BLOOM*
May 28, 2013
Abstract.
For Y a subset of the complex plane, a β ensembleis a sequence of probability measures P rob n,β,Q on Y n for n =1 , , . . . depending on a positive real parameter β and a real-valuedcontinuous function Q on Y. We consider the associated sequenceof probability measures on Y where the probability of a subset W of Y is given by the probability that at least one coordinate of Y n belongs to W. With appropriate restrictions on
Y, Q we prove alarge deviation principle for this sequence of probability measures.This extends a result of Borot-Guionnet to subsets of the complexplane and to β ensembles defined with measures using a Bernstein-Markov condition. introduction β ensembles are generalizations of the joint probability distributionsof the eigenvalues of the classical matrix ensembles. They are defined asfollows: let Y be a closed subset of C , Q a real-valued continuous func-tion on Y and β > . Consider the family of probability distributions
P rob n,β,Q for n = 1 , , ... defined on Y n by:(1.1) P rob n,β,Q = A n,β,Q ( z ) Z n,β,Q dτ ( z )where(1.2) A n,β,Q ( z ) := | D ( z , . . . , z n ) | β exp( − n [ Q ( z ) + . . . + Q ( z n )]) ,D ( z , . . . , z n ) = Y ≤ i Key words and phrases. equilibrium measure.*supported by an NSERC of Canada grant. denotes the Vandermonde determinant, and the normalizing constants Z n,β,Q are given by:(1.3) Z n,β,Q ( Y ) = Z n,β,Q := Z Y n A n,β,Q ( z ) dτ ( z ) . Here dτ ( z ) = dτ ( z ) . . . dτ ( z n )and τ is an appropriate measure on Y. In particular, the support of τ is Y. We assume that R = 2 Q/β is of superlogarithmic (see (2.1))growth if Y is unbounded. The existence of the integrals in the case of Y unbounded is dealt with in the course of proving Theorem 7.3.In this paper we will assume the measure τ satisfies a weightedBernstein-Markov inequality (see section 3 and Hypothesis 3.9). Lebesguemeasure in one or two dimensions satisfies Hypothesis 3.9 but, in addi-tion, there are more general measures which also satisfy Hypothesis 3.9.In fact there are discrete measures (i.e a countable linear combinationof Dirac measures) which satisfy the hypothesis.For Y = R , dτ = dx (Lebesgue measure) , Q ( x ) = x / β =1 , , x = ( x , . . . , x n ) are referred to as eigenvalues.The 2-dimensional version of these probability distributions occursin the study of the Coulomb gas model ([11], [12]). In this modelthe parameter β corresponds to the inverse temperature, 2 Q to theconfining potential, and the coordinates of a point are the positions ofparticles.The two-dimensional version also occurs, for β = 2, as the distribu-tion of eigenvalues of random normal matrices ([12]).These ensembles have been extensively studied, primarily in the case Y = R or C and dτ as Lebesgue measure (see [1], [11], [12], [13] andthe references given there). In particular, it is known that the nor-malized counting measure of a random point, i.e. the random measure n P nj =1 δ ( z j ), where δ denotes the Dirac measure, converges, almostsurely, weak ∗ , to a non-random measure with compact support - theweighted equilibrium measure (see (2.3)) . Furthermore a large devia-tion principle for the normalized counting measure of a random pointis known [1]. The large deviation principle has speed n and implies ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 3 that the probability of finding a positive proportion of the eigenval-ues/particles/coordinates outside a neighbourhood of the support ofthe weighted equilibrium measure is asymptotic to e − n c for some c > . In this paper we are concerned with the behaviour of a single eigen-value/particle/coordinate. We will establish a large deviation principlefor the coordinate z ∈ Y under the probability distributions (1.1).That is we will prove a large deviation principle for the countable fam-ily of probability distributions on Y given by, for W an open subset of Y (1.4) ψ n ( W ) = P rob n,β,Q { z ∈ W } = 1 Z n,β,Q Z W Z Y n − A n,β,Q ( z ) dτ ( z )for n = 1 , , . . . . The main results of this paper are Theorems 6.2 and 7.3 which showthat the sequence of probability measures ψ n satisfies a large deviationprinciple with speed n and rate function(1.5) J Y,β,Q ( z ) = 2 Q ( z ) − β ( Z Y log | z − t | dµ Y,β,Q + ρ )where µ Y,β,Q is the weighted equilibrium measure (see (2.3)) and ρ is aconstant (see(2.5)).This l.d.p. implies that the probability of finding an eigenvalue/particle/coordinate outside a neighbourhood of the support of the weightedequilibrium measure is asymptotic to e − nc for some c > . Theorem 6.2 handles the case when Y is compact and Theorem 7.3the unbounded case. Theorem 6.2 is valid under the following hypothe-ses. (We let S R denote the support of the weighted equilibrium measureand S ∗ R the points where J n,β,Q ( z ) = 0)(1) Y is a regular set (in the sense of potential theory.)(2) (Hypothesis 3.9) τ satisfies the weighted BM inequality on anycompact neighbourhood of S ∗ R . (3) (Hypothesis 6.1) S ∗ R = S R . Now (1) ensures that the weighted Green function (see (2.2)) is con-tinuous. (2) is always satisfied by Lebesgue measure on R or C (seesection 3). (3) is the assumption that the rate function is strictly pos-itive outside S R . In [9] the corresponding assumption is referred to ascontrol of large deviation.The unbounded case, Theorem 7.3, requires additional assumptions.Theorem 7.3 is valid under the following assumptions: Condition (1) THOMAS BLOOM* above, is replaced by the hypothesis that the intersection of Y withsufficiently large discs centre the origin is regular. (2) and (3) abovemust hold. The additional conditions are on the growth of τ and R =2 Q/β. Namely:(4) (see(2.1)) For some b > , lim | z |→∞ ( R ( z ) − (1 + b ) log | z | ) = + ∞ . (5) (Hypothesis 7.1) For some a > , R Y dτ / | z | a < + ∞ . A key step is theorem 5.1 where we prove a result on the asymptoticsof normalizing constants, proved in [9] for subsets of R and used previ-ously as an assumption in [1]. Our methods use polynomial estimatesand potential theory but not the large deviation principle for the nor-malized counting measures of a random point. We will specifically useweighted potential theory (see [19]), since the probability distributionsgiven by (1.1) are, in each variable, a power of the absolute value of aweighted polynomial (see section 2).These results extend a result of [9] to appropriate subsets of theplane and, in addition, the measure τ used to define the ensemblecan be more general than Lebesgue measure: it need only satisfy theBernstein-Markov condition given by Hypothesis 3.9. The rate functionis independent of τ as long as τ satisfies Hypothesis 3.9.We let C, c denote positive constants which may vary from line toline. All measures are positive Borel measures.2. polynomial estimates We will list some basic results of weighted potential theory (see [19]).Let Y be a closed set in the plane and R be a continuous (real-valued)function on Y . If Y is unbounded , R is assumed to be superlogarithmici.e., for some b > | z |→∞ ( R ( z ) − (1 + b ) log | z | ) = + ∞ . For r > Y r := { z ∈ Y || z | ≤ r } . We will assume that Y r is aregular set (in the sense of potential theory) for all r sufficiently large.We recall that a compact set is regular if it is regular for the exteriorDirichlet problem [17] or equivalently if the unweighted Green function(i.e. R ≡ C . The weighted Green function of Y with respect to R ([19], AppendixB) is denoted by V Y,R . It is defined by(2.2) V Y,R ( z ) = sup { u ( z ) | u is subharmonic on C , u ≤ R on Y and, ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 5 u ≤ log + | z | + C, where C is a constant depending on u } .V Y r ,R is continuous for r sufficiently large ([18], prop 2.16) since Y r wasassumed to be regular for r sufficiently large and any regular compactset in the plane is locally regular ([17]). Thus V Y,R is continuous since V Y,R = V Y r ,R for r sufficiently large ([19], Appendix B, lemma 2.2).The logarithmic capacity c ( E ) of a Borel subset E ⊂ C is defined bylog c ( E ) = sup ν { Z Z log | z − t | dν ( z ) dν ( t ) } over all Borel probability measures ν with compact support in E. Regular sets are of positive capacity and so Y is of positive capacity.The function e − R is thus strictly positive on a set of positive capacityand so is an admissible weight in the terminology of [19]. Thus ChapterI, Theorem 1.3 of [19] holds.The weighted equilibrium measure is denoted µ Y,R . It has compactsupport and is the unique minimizer of(2.3) E ( ν ) = − Z Z log | z − t | dν ( z ) dν ( t ) + 2 Z R ( z ) dν ( z )over all measures ν ∈ M ( Y ) where M ( Y ) denotes the probabilitymeasures on Y ([19], Chapter I, Theorem 1.3).Now, by ([19], Appendix B, Lemma 2.4)(2.4) V Y,R ( z ) = Z Y log | z − t | dµ Y,R ( t ) + ρ where ρ is the Robin constant given by(2.5) ρ = lim | z |→∞ ( V Y,R ( z ) − log | z | ) . It is also known that E ( µ Y,R ) > −∞ and ([19], Chapter I, Theorem3(d)) ρ = E ( µ Y,R ) − Z Y R ( z ) µ Y,R . We let S ∗ R := { z ∈ Y | V Y,R = R } and S R := supp( µ Y,R ). S R and S ∗ R are compact and in general S R ⊂ S ∗ R ([19], Chapter I, Theorem 1.3(e)). S R and S ∗ R depend on Y although our notation does not explicitly indi-cate this. However, for r sufficiently large S R and S ∗ R are independentof Y r . THOMAS BLOOM* From (2.4) we have ([19], Chapter I, Theorem 1.3(f))(2.6) R ( z ) = Z Y log | z − t | dµ Y,R ( t ) + ρ for z ∈ S R . We let P n denote the space of polynomials in the single variable z of degree ≤ n and for p ∈ P n we refer to e − nR ( z ) p ( z ) as a weightedpolynomial of degree n (the weight is the positive continuous function e − R ( z ) ) . Now for β > R ( z ) = 2 β Q ( z ) . Note that A n,β,Q ( z ) is, in each variable, of the form | e − nR ( z ) p ( z ) | β = | e − nQ p ( z ) | β and is thus the absolute value of a weighted polynomial to the β power.For this reason weighted potential theory can be used in the study of β ensembles.Equations (2.8) and (2.9) below are known estimates for weightedpolynomials.By ([19], Chapter III, Corollary 2.6) the sup norm of a weightedpolynomial is assumed on S R . That is(2.8) || e − nR p ( z ) || Y = || e − nR ( z ) p ( z ) || S R for all p ∈ P n . By ([19], Chapter I, Theorem 3.6) we have, for p a monic polynomialof degree n (2.9) || e − nR ( z ) p ( z ) || S R ≥ e − nρ . ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 7 For G a subset of M ( Y ) we will use the notation˜ G n = { z = ( z , . . . , z n ) ∈ Y n (cid:12)(cid:12)(cid:12) n n X j =1 δ ( z j ) ∈ G } . Here δ denotes the Dirac delta measure.We now restrict to the case that Y is compact and we will use thenotation K in place of Y. Lemma 2.1. Given ǫ > , there is a neighbourhood G of µ K,R in M ( K ) (with the weak* topology) such that for ( z , . . . , z n ) ∈ ˜ G n we have || e − nR ( t ) ( t − z ) . . . ( t − z n ) || K ≤ e − n ( ρ − ǫ ) . Proof. The proof will be by contradiction. If not, for some ǫ > 0, nosuch G exists. Thus there exists a sequence of n s -tuples ( z s , . . . , z sn s )for s = 1 , , . . . with(2.10) lim s n s n s X j =1 δ ( z sj ) = µ K,R weak*, but, using (2.8),(2.11) || e − n s R ( t ) ( t − z s ) . . . ( t − z sn s ) || S R ≥ e − n s ρ e n s ǫ . Taking logarithms this may be rewritten as:(2.12) (cid:13)(cid:13)(cid:13) − R ( t ) + 1 n s n s X j =1 log | t − z sj | (cid:13)(cid:13)(cid:13) S R ≥ − ρ + ǫ. It follows from (2.10) that, using ([19], Chapter 0, Theorem 1.4), wehave(2.13) lim sup s n s n s X j =1 log | t − z sj | ≤ Z log | t − ξ | dµ K,R ( ξ ) , since ξ → log | t − ξ | is uppersemicontinuous. Now, (2.13) is a pointwise lim sup and to obtain a contradiction we will require a uniform lim supfor t ∈ S R . THOMAS BLOOM* Since ξ → log | t − ξ | is subharmonic, by Hartogs’ lemma, ([15], The-orem 2.6.4) and (2.6) we have(2.14) lim sup s n s n s X j =1 log | t − z sj | ≤ R ( t ) − ρ + ǫ/ , uniformly on S R . That is, for s sufficiently large,(2.15) (cid:13)(cid:13)(cid:13) − R ( t ) + 1 n s n s X j =1 log | t − z sj | (cid:13)(cid:13)(cid:13) S R ≤ − ρ + ǫ/ , which contradicts (2.12). (cid:3) The following is a corollary to the proof of Lemma 2.1 and will beused in the proof of Theorem 6.2. Corollary 2.2. Given ǫ > , there is a neighborhood G of µ K,R in M ( K ) (with the weak* topology) such that for ( z , . . . , z n ) ∈ ˜ G n − and w ∈ K then (2.16) ( 1 n − n X j =2 log | w − z j | − Z log | w − t | dµ K,R ( t )) ≤ ǫ. Bernstein-Markov inequalities Although we ultimately will only need Bernstein-Markov inequalitiesfor subsets of C we will consider these inequalities in C N as the C case is used in the proofs. Recall that all measures are positive Borelmeasures. Definition 3.1. Let τ be a measure on a compact set K ⊂ C N . Wesay τ satisfies the Bernstein-Markov (BM) inequality if, for all ǫ > C > n, p ) such that(3.1) || p || K ≤ C (1 + ǫ ) n Z K | p | dτ, for all p ∈ P Nn , where here P Nn denotes the holomorphic polynomials oftotal degree ≤ n on C N .. ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 9 Definition 3.2. Let τ be a measure on a compact set K ⊂ C N and R a real-valued continuous function on K. We say τ satisfies the weightedBernstein-Markov (BM) inequality for the weight e − R if, for all ǫ > C > n, p ) such that(3.2) || e − nR p || K ≤ C (1 + ǫ ) n Z K e − nR | p | dτ, for all p ∈ P Nn . We say that τ satisfies a strong BM inequality if it satisfies the weightedBM inequality for all weights e − R . It is known ([20], proof of Theorem 3.4.3 ) that if τ satisfies (3.1)or (3.2) then it also satisfies an L β version of that inequality (with,possibly, a different constant C ). For example, from (3.2)(3.3) || e − nR p || K ≤ C (1 + ǫ ) n ( Z K ( e − nR | p | ) β dτ ) β . The following lemma will be used in the proof of Theorem 5.1. Lemma 3.3. Let K ⊂ C be regular, R a continuous real-valued contin-uous function on K and τ be a measure on K satisfying the weightedBM inequality for the weight e − R . Then for all monic polynomials ofdegree n (3.4) Z K | e − nQ p β | dτ ≥ C (1 + ǫ ) − nβ e − nρβ . Proof. Recall that R = 2 Q/β and combine (3.2) with (2.8) and (2.9). (cid:3) Conditions on a measure to satisfy the BM inequality for compactsets K ⊂ C are extensively studied in [20]. The measures termed regular in [20] coincide with measures satisfying the BM inequality on regularcompact sets but the BM inequality is a more stringent condition ingeneral (see [20], example 3.5.3). Remark 3.4. Suppose that τ satisfies the weighted BM inequality for e − R on S R . Then it follows from (3.2) and (2.8) that τ satisfies theweighted BM inequality for e − R on any compact set K ⊃ S R . Hence any measure τ which satisfies the weighted BM inequality for e − R on S R will satisfy Hypothesis 3.9 below i.e. τ satisfies the weighted BM inequality on any compact neighbourhood of S R . Now for every compact set K ⊂ C there exists a discrete measure ( i.e. a countablelinear combination of Dirac δ measures ) on K which satisfies the strongBM inequality (see [5], Corollary 3.8 - the construction given there is,in fact, valid for every compact set). Applying this construction to S R gives a discrete measure which satisfies Hypothesis 3.9. We will showthat Lebesgue measure on R or C also satisfies Hypothesis 3.9.The next proposition gives a convenient sufficient (but by no meansnecessary) condition for a measure τ to satisfy the BM inequality on aregular compact set K ⊂ C N (for the definition of regular compact setin C N , N > Proposition 3.5. ( [7] , Theorem 2.2) Let K ⊂ C N be compact andregular. Let τ be a measure on K such that for some T > and each z ∈ K there exists r ( z ) > such that (3.5) τ { B ( z , r ) } ≥ r T for r ≤ r ( z ) where B ( z , r ) denotes the ball centre z , radius r. Then τ satisfies theBM inequality on K. We also have: Proposition 3.6. ( [3] , Theorem 3.2) Let K ⊂ R N ⊂ C N . Then if τ satisfies the BM inequality on K it satisfies the strong BM inequalityon K. Corollary 3.7. Let K be the closure of its interior in C and have C boundary. Then Lebesgue planar measure satisfies the strong BMinequality on K. Proof. Consider K ⊂ C ∼ = R ⊂ C . Now, K is regular as a sub-set of C (this follows, for example from the accessibility condition ofPlesniak,[15], Chapter 5) and any measure satisfying a strong BM in-equality on K as a subset of C satisfies a strong BM inequality on K as a subset of C . Thus the result follows from Propositions 3.5 and3.6. (cid:3) Corollary 3.8. Let K ⊂ R be a finite union of disjoint closed inter-vals.Then Lebesgue measure on R satisfies the strong BM inequality on K. ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 11 Proof. K is regular as a subset of C ([20]) so it follows from Proposi-tions 3.5 and 3.6 that Lebesgue measure on R satisfies the strong BMinequality on K. (cid:3) We will use the following hypothesis in the next lemma: Hypothesis 3.9. The measure τ satisfies the weighted BM inequalityfor e − R on any compact neighbourhood of S ∗ R . Remark 3.10. Every compact set K ⊂ C has a neighbourhood ba-sis of compact sets which are the closure of their interiors and have C boundary. Thus by Corollary 3.7, Lebesgue measure on C satis-fies Hypothesis 3.9. Similarly, since every compact set K ⊂ R has aneighbourhood basis of compact sets which are a finite disjoint unionof closed intervals, so by Corollary 3.8, Lebesgue measure on R satisfiesHypothesis 3.9. Lemma 3.11. Let K ⊂ C be regular, R a real-valued continuous func-tion on K and τ be a measure on K satisfying Hypothesis 3.9. Let N be a compact neighbourhood of S ∗ R . Then there is a constant c > (independent of n, p ) such that: Z K | e − nR p | β dτ ≤ (1 + O ( e − nc )) Z N | e − nR p | β dτ for all p ∈ P n . Proof. We first normalize the polynomial so that || e − nR p || S R = 1 . Toprove the theorem it will suffice to show that(3.6) Z K \ N | e − nR ( z ) p ( z ) | β dτ ≤ Ce − nc , for some constants C, c > n (cid:16) Z K | e − nR p | β dτ (cid:17) /n ≥ . We will use the estimate ([19], Appendix B, Theorem 2.6 (ii))(3.8) | e − nR ( z ) p ( z ) | ≤ || e − nR ( z ) p ( z ) || S R exp( n ( V K,R ( z ) − R ( z ))for z ∈ K. Since N is a compact neighbourhood of S ∗ R , for z ∈ K \ N , and someconstant b > V K,R ( z ) − R ( z ) ≤ − b < so | e − nR ( z ) p ( z ) | ≤ e − nb for z ∈ K \ N. Thus,(3.10) Z K \ N | e − nR ( z ) p ( z ) | β dτ ≤ Ce − nbβ for constants C, b > . Now,(3.11) Z K | e − nR ( z ) p ( z ) | β dτ ≥ Z N | e − nR ( z ) p ( z ) | β dτ and since τ satisfies the weighted BM condition on N for e − R the righthand side in (3.11) is, for any ǫ > C > ≥ || e − nR p || βS R Ce − ǫn = Ce − ǫn , establishing (3.7) and the result. (cid:3) Theorem 7.2 establishes a version of this lemma for unbounded sets.4. Johansson large deviation We will not use the l.d.p. for the normalized counting measure of arandom point but a weaker result whose utility was shown by Johansson[14].Consider(4.1) A n,β,Q ( z ) := | D ( z , . . . , z n ) | β exp( − n [ Q ( z ) + . . . + Q ( z n )])where β > D denotes the Vandermonde determinant D ( z , . . . , z n ) = Y ≤ i We obtain a collection of probability measures which we refer to as a β ensemble on K. We let, for ν ∈ M ( K )(4.4) E β ( ν ) = − β/ Z Z log | x − y | dν ( x ) dν ( y ) + 2 Z Q ( x ) dν ( x ) . Then(4.5) E β ( ν ) = β E ( ν ) . Thus the unique minimizer in M ( K ) of E β ( ν ) is µ K,R for which we willalso use the notation µ K,β,Q . The functional E β is a special case of the functionals E Q ( µ ) studiedin [8] and it is analogous to the functionals E ( ν ) and E Q ( ν ) consideredin ([2],equations (2.9) and (5.4)).The following three propositions aretherefore special cases of results in [8] and the proofs are also analogousto proofs in [2]. Proposition 4.1. lim n n log Z n,β,Q = − E β ( µ K,β,Q ) = lim n n log sup K n A n,β,Q ( z ) Proof. See [8],Proposition 4.15 and Proposition 3.2 or [2], Theorem 3.9and Corollary 3.6. (cid:3) We note that the two quantities in the equality on the right of Propo-sition 4.1 do not depend on the measure τ. Let log γ := − E β ( µ K,β,Q ) . Then γ > η , 0 < η < γ be given.We define B Qη,n,β := { z ∈ K n | A n n,β,Q ( z ) ≤ γ − η } . Then we have the following result, which we refer to as a Johanssonlarge deviation result: Proposition 4.2. P rob n,β,Q ( B Qη,n,β ) = 1 Z n,β,Q Z B Qη,n,β A n,β,Q ( z ) dτ ( z ) ≤ (1 − η γ ) n τ ( K ) n for all n sufficiently large.Proof. See [8], Proposition 4.15 or [2], Theorem 4.1. (cid:3) Proposition 4.3. Let G be a neighbourhood of µ K,β,Q in M ( K ) . Then Z n,β,Q Z K n \ ˜ G n A n,β,Q ( z ) dτ ≤ O ( e − cn ) for some c > . Proof. It follows (see [8], claim (6.16) or Proposition 7.3 of [2]) that forsome η > B Qη,n,β ⊃ K n \ ˜ G n for all n sufficiently large, and so the result then follows from Proposi-tion 4.2. (cid:3) the normalizing constants Proposition 4.1 gives an asymptotic result for the normalizing con-stants Z n,β,Q . Note that if we have a sequence of continuous func-tions { Q n } converging uniformly to Q on K , then n log Z n,β,Q and n log Z n,β,Q n have the same limit, and in particular, this is true for n log Z n,β, nQn − . We will, however, need a sharper result, namely: Theorem 5.1. Let K be a regular compact subset of C , Q a real-valuedcontinuous function on K and τ a measure on K satisfying the weightedBM inequality for e − R . Then lim n →∞ (cid:16) Z n,β,Q Z n − ,β, nQn − (cid:17) n = e − ρβ . Proof. We will first prove that(5.1) lim inf n →∞ (cid:16) Z n,β,Q Z n − ,β, nQn − (cid:17) n ≥ e − ρβ . Now Z n,β,Q = R K n A n,β,Q ( z ) dτ ( z ) . We regard A n,β,Q ( z ) as a function of z and we apply Lemma 3.3 to the integral in the z variable to obtain(5.2) Z n,β,Q ≥ C (1+ ǫ ) − nβ e − nρβ Z K n | D ( z , . . . , z n ) | β e − n [ Q ( z )+ ... + Q ( z n )] dτ ( z ) . . . dτ ( z n )= C (1 + ǫ ) − nβ e − nρβ Z n − ,β, nQn − . Since ǫ > ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 15 To complete the proof we must show(5.3) lim sup n →∞ (cid:16) Z n,β,Q Z n − ,β, nQn − (cid:17) n ≤ e − ρβ . Let ˜ G n − be a subset of K n − determined as follows: Given ǫ > G of µ K,β,Q so that Lemma 2.1 holds. Recallthat ˜ G n − = { ( z , . . . , z n ) ∈ K n − | n − n X j =2 δ ( z j ) ∈ G } . We write the integral for Z n,β,Q as a sum of two integrals Z n,β,Q = I + I where I = Z K Z K n − \ ˜ G n − A n,β,Q ( z ) dτ ( z )and I = Z K Z ˜ G n − A n,β,Q ( z ) dτ ( z ) . Now,(5.4) I = Z K n Y j =2 | z − z j | β e − nQ ( z ) dτ ( z ) × Z K n − \ ˜ G n − | D ( z , . . . , z n ) | β e − n [ Q ( z )+ ... + Q ( z n )] dτ ( z ) . . . dτ ( z n ) . Since K is compact, the first factor is O ( C n ) for some C > . Theintegrand in the second factor differs from A n − ,β,Q ( z , . . . , z n ) by afactor of O ( C n ) for some C > O ( e − cn ) Z n − ,β,Q by Proposition 4.3. Since Z n − ,β,Q = O ( C n ) Z n − ,β, nQn − , we may conclude that I = O ( e − cn ) Z n − ,β, nQn − . Also(5.5) I = Z K n Y j =2 | z − z j | β e − nQ ( z ) dτ ( z ) × Z ˜ G n − | D ( z , . . . , z n ) | β e − n [ Q ( z )+ ... + Q ( z n )] dτ ( z ) . . . dτ ( z n ) . We will need a more precise estimate on the first factor than the oneused in (5.4). Since we are integrating over ˜ G n − we may use Lemma2.1 on the first factor to see that it is ≤ Ce − nβ ( ρ − ǫ ) . The second factoris ≤ Z n − ,β, nQn − since ˜ G n − is a subset of K n − . Hence, given any ǫ > c, C > I + I ≤ Ce − nβ ( ρ − ǫ ) Z n − ,β, nQn − + O ( e − cn ) Z n − ,β, nQn − and (5.3) follows. (cid:3) large deviation Given a separable, complete metric space X, a sequence of proba-bility measures { σ n } on X is said to satisfy a large deviation princi-ple with speed n and rate function J ( x ) if J : X → [0 , ∞ ] is lower-semicontinuous, { x ∈ X | J ( x ) ≤ l } is compact for l ≥ F ⊂ X we havelim sup n n log σ n ( F ) ≤ − inf x ∈ F J ( x ) . (ii) For all open sets G ⊂ X we havelim inf n n log σ n ( G ) ≥ − inf x ∈ G J ( x ) . If X is compact by ([10], Theorem 4.1.11) to establish the l.d.p. itsuffices to show that, for all x ∈ X (6.1) − J ( x ) = lim ǫ → lim n n log σ n ( B ( x, ǫ ))where B ( x, ǫ ) is the ball centre x , radius ǫ. If X is non-compact, toestablish the l.d.p. there is an additional condition required, termed exponential tightness, namely: For all r > X r ⊂ X with σ n ( X r ) ≤ − r. Let K ⊂ C be a regular compact set and Q a real - valued continuousfunction on K . Let τ be a measure on K which satisfies the weightedBM inequality for e − R (recall that R = 2 Q/β ). Given the probabilitymeasures P rob n,β,Q forming a β ensemble on K , we consider the count-able family of probability measures { ψ n } , n = 1 , , ... on K given asfollows: For W an open subset of K ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 17 (6.2) ψ n ( W ) = P rob n,β,Q { z ∈ W } = 1 Z n,β,Q Z W Z K n − A n,β,Q ( z ) dτ ( z ) . To prove the l.d.p., we will assume that τ satisfies Hypothesis 3.9.We will also use an additional assumption. Hypothesis 6.1. (6.3) S R = S ∗ R . Now, to prove the l.d.p. we will use equation (6.1) and so we willhave to estimate inf { W | z ∈ W } lim n n log ψ n ( W ) . We will be able to do so for z ∈ K \ S ∗ R and z ∈ S R . Thus underHypothesis 6.1 the l.d.p. will be complete.For β ensembles on R , τ the Lebesgue measure on R and Q realanalytic, then results of [16] show that Hypothesis 6.1 holds for all butat most countably many values of β. A class of examples where Hypothesis 6.1 holds are provided by thecomputations in ([19], Chapter IV, section 6). We suppose R ( z ) isinvariant under rotations of the plane, that R ( t ) is differentiable on(0 , ∞ ) and satisfies tR ′ ( t ) and R ( t ) are increasing on (0 , ∞ ). Then S R is a disc of radius T given by the solution of T R ′ ( T ) = 1 . If we take K a disc of radius > T , then V K,R ( z ) = log | z | + R ( T ) − log T while R ( z ) > V K,R ( z ) for | z | > T so that S ∗ R = S R . R ( z ) = | z | provides aspecific example.It is simple to construct examples where Hypothesis 6.1 does nothold. For example, let K be a large disc with S ∗ R ⊂ interior( K ) . Re-placing R by R ′ = V K,R then V K,R ′ = V K,R so S R ′ = S R but S ∗ R ′ = K. Theorem 6.2. Let K be a regular, compact subset of C , Q a real-valuedcontinuous function on K and τ a measure with supp( τ ) = K andsatisfying Hypotheses 3.9 and 6.1. Let P rob n,β,Q be a β ensemble on K. Then the sequence of measures ψ n , given by ψ n ( W ) = P rob n,β,Q { z ∈ W } for W an open subset of K satisfies a l.d.p. with speed n and ratefunction (6.4) J K,β,Q ( z ) = β ( R ( z ) − V K,R ( z )) = 2 Q ( z ) − βV K,R ( z ) . Proof. Using (2.4) we have J K,β,Q ( z ) = 2 Q ( z ) − β ( Z K log | z − t | dµ K,β,Q ( t ) + ρ ) . Note that J K,β,Q ( z ) = 0 for z ∈ S R . Also since V K,R is continuous, the function(6.5) z → Z K log | z − t | dµ K,β,Q ( t )is continuous and so is the rate function.We will use equation (6.1). Let W be an open subset of K . We willestimate the probability that a point w ∈ W. By definition,(6.6) P rob n,β,Q { w ∈ W } = 1 Z n,β,Q Z W Z K n − A n,β,Q ( w, z , ..., z n ) dτ. We will separately estimate lim sup n and lim inf n of1 /n log( P rob n,β,Q { w ∈ W } )We begin with the lim sup . We write the above integral as a sum oftwo integrals (where G is an open neighbourhood of µ K,β,Q ⊂ M ( K )which is to be specified).1 Z n,β,Q Z W Z K n − A n,β,Q ( w, z , ...z n ) dτ = H + H .H = 1 Z n,β,Q Z W Z K n − \ ˜ G n − A n,β,Q ( w, z , ...z n ) dτ and H = 1 Z n,β,Q Z W Z ˜ G n − A n,β,Q ( w, z , ...z n ) dτ. Similar to the estimate for I , (see (5.4)) we have H = O ( e − cn ) . To estimate H we now proceed as in [1]. Let(6.7) h n := Z n,β,Q Z n − ,β, nQn − . Then(6.8) H = 1 h n Z n − ,β, nQn − Z W Z ˜ G n − A n,β,Q ( w, z , ...z n ) dτ. ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 19 Now, given ǫ > 0, using Corollary 2.2, let G be a neighbourhood of µ K,β,Q ⊂ M ( K ) so that for w ∈ W and ( z , . . . , z n ) ∈ ˜ G n − then(6.9) β (cid:16) n − n X j =2 log | w − z j | − Z log | w − t | dµ K,β,Q ( t ) (cid:17) ≤ ǫ. Taking exponentials(6.10) n Y j =2 | w − z j | β ≤ e ( n − ǫ + β R log | w − t | dµ K,β,Q ( t )) . Also, since ˜ G n − ⊂ K n − (6.11)1 Z n − ,β, nQn − Z ˜ G n − | D ( z , . . . , z n ) | β e − n [ Q ( z )+ ... + Q ( z n )] dτ ( z ) . . . dτ ( z n ) ≤ . Using Theorem 5.1 to estimate h n , the inequalities in (6.10) and(6.11), and recalling that τ ( W ) = 0 since the support of τ is K , wehave(6.12) lim sup n /n log( P rob n,β,Q { w ∈ W } ) ≤ βρ + sup w ∈ W ( β Z K log | w − t | dµ K,β,Q − Q ( w )) . and using (6.5) we have(6.13) inf { W | z ∈ W } lim sup n /n log( P rob n,β,Q { w ∈ W } ) ≤ βρ + β Z K log | z − t | dµ K,β,Q − Q ( z ) . Now we must deal with the lim inf . First we consider z / ∈ S R . Let W be a neighbourhood of z with W ∩ S R = ∅ , and let N be a compactneighbourhood of S R such that N ∩ W = ∅ . Now A n,β,Q ( z ) is in eachvariable of the form e − nQ | p | β for a polynomial p ∈ P n . Hence byrepeated use of Lemma 3.11 we have,(6.14) Z K n A n,β,Q ( z ) dτ ( z ) ≤ (1 + O ( e − cn )) Z N n A n,β,Q ( z ) dτ ( z ) , or(6.15) Z n,β,Q ( K ) ≤ (1 + O ( e − cn )) Z n,β,Q ( N ) where K and N indicate the sets on which the normalizing constantsare evaluated. Similarly,(6.16) Z n − ,β, nQn − ( K ) ≤ (1 + O ( e − cn )) Z n − ,β, nQn − ( N ) . Now 1 Z n,β,Q ( K ) Z W Z K n − A n,β,Q ( w, z , ...z n ) dτ ≥ Z n,β,Q ( K ) Z W Z N n − A n,β,Q ( w, z , ...z n ) dτ and using (6.14), to obtain the lower bound it suffices to estimate:1 Z n,β,Q ( N ) Z W Z N n − A n,β,Q ( w, z , ...z n ) dτ =1 h n ( N ) Z n − ,β, nQn − ( N ) Z W Z N n − A n,β,Q ( w, z , ...z n ) dτ. Given ǫ > F be a neighbourhood of µ K,β,Q in M ( N ) such that for w ∈ W and ( z , . . . , z n ) ∈ ˜ F n − we have,(6.17) − ǫ ≤ β (cid:16) n − n X j =2 log | w − z j | − Z log | w − t | dµ K,β,Q ( t ) (cid:17) . Such an F exists since log | w − t | is continuous for t ∈ N and w ∈ W. Taking exponentials(6.18) n Y j =2 | w − z j | β ≥ e ( n − − ǫ + β R log | w − t | dµ K,β,Q ( t )) . Now we have (see (5.4))(6.19) 1 Z n − ,β, nQn − ( N ) Z N n − \ ˜ F n − | D ( z , . . . , z n ) | β e − n [ Q ( z )+ ... + Q ( z n )] dτ ≤ O ( e − cn ) . So(6.20)1 − O ( e − cn ) ≤ Z n − ,β, nQn − ( N ) Z ˜ F n − | D ( z , . . . , z n ) | β e − n [ Q ( z )+ ... + Q ( z n )] dτ ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 21 Using the inequalities in (6.18) and (6.20), and the fact that τ ( W ) = 0 , we have,(6.21) lim inf n /n log( P rob n,β,Q { w ∈ W } ) ≥ βρ + inf w ∈ W ( β Z K log | w − t | dµ K,β,Q − Q ( w )) . By (6.5) it follows that(6.22) inf { W | z ∈ W } lim n n log( P rob n,β,Q { w ∈ W } )= β ( ρ + Z K log | z − t | dµ K,β,Q ) − Q ( z ) . To complete the l.d.p. we must consider the case when z ∈ S R andto do so we must estimate P rob n,β,Q { w ∈ W } when W ∩ S R = ∅ . Infact, we will show that, in this case,(6.23) lim n n log( P rob n,β,Q { w ∈ W } ) = 0 . Now, nP rob n,β,Q { w ∈ W } ≥ P rob n,β,Q { at least one of w, z , . . . , z n ∈ W } = 1 − P rob n,β,Q { each of w, z , . . . , z n ∈ K \ W } . Since W ∩ S R = ∅ the support of the weighted equilibrium mea-sure for R on K \ W cannot be S R . This implies, using Proposi-tion 4.1 and the minimizing property of the equilibrium measure, thatlim sup n sup K \ W A n,β,Q ( z ) n ≤ γ − η, for some η > . (Recall that γ = lim sup n sup K A n,β,Q ( z ) n . ) Then one can use Proposition 4.2 toobtain(6.24) P rob n,β,Q { w ∈ W } ≥ n (1 − O ( e − cn ))and (6.23) follows. (cid:3) the unbounded case Let Y be a closed, unbounded subset of C . Let R be a continuous,real - valued, superlogarithmic function on Y. That is, for some b > , lim | z |→∞ ( R ( z ) − (1 + b ) log | z | ) = + ∞ . For r > Y r =: { z ∈ Y || z | ≤ r } and we assume Y r is regu-lar for r sufficiently large. Also, for r sufficiently large V Y,R = V Y r ,R ([19],Appendix B, Lemma 2.2) and S ∗ R ⊂ Y r . We will also denote theequilibrium measure µ Y,R by µ Y,β,Q . We will extend the l.d.p. (Theorem 6.2) from the compact case to theunbounded case. We will need an additional hypothesis on the growthof the measure τ. Hypothesis 7.1. τ is a locally finite measure on Y satisfying: (7.1) For some a > , we have Z Y dτ / | z | a < + ∞ We note that Lebesgue measure on R or C satisfies Hypothesis 7.1.The following theorem will extend Lemma 3.11 to the unboundedcase. Both Lemma 3.11 and Theorem 7.2 are based on ([19], Chap-ter III, Theorem 6.1). They shows that the L β norm of a weightedpolynomial ”lives” on S ∗ R . Theorem 7.2. Let Y be a closed, unbounded subset of C with Y r regularfor r sufficiently large. Let R be a superlogarithmic real-valued functionon Y. Let β > and let N ⊂ Y be a compact neighbourhood of S ∗ R . Let τ be a measure on Y such that Hypotheses 3.9 and 7.1 are satisfied.Then there is a constant c > (independent of n, p ) such that: Z Y | e − nR p | β dτ ≤ (1 + O ( e − nc )) Z N | e − nR p | β dτ for all p ∈ P n . Proof. We need only consider the case N = Y r for r large and then theresult will follow by using Lemma 3.11.We first normalize the polynomial so that || e − nR p || S R = 1 . To provethe theorem it will suffice to show that(7.2) Z Y \ Y r | e − nR ( z ) p ( z ) | β dτ ≤ Ce − nc , ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 23 for some constants C, c > n (cid:16) Z Y | e − nR p | β dτ (cid:17) /n ≥ . We will use the estimate ([19] , Appendix B, Theorem 2.6 (ii))(7.4) | e − nR ( z ) p ( z ) | ≤ || e − nR ( z ) p ( z ) || S R exp( n ( V Y,R ( z ) − R ( z ))for z ∈ Y. Using (2.1) we have, since V Y,R ≤ log | z | + C for | z | large(7.5) V Y,R ( z ) − R ( z ) ≤ − b | z | for | z | large .Hence, for r large(7.6) Z Y \ Y r | e − nR ( z ) p ( z ) | β dτ ≤ C Z Y \ Y r dτ | z | nbβ ≤ Cr − nbβ r a Z Y \ Y r dτ | z | a ≤ Ce − nc . for constants C, c > n sufficiently large.Now,(7.7) Z Y | e − nR ( z ) p ( z ) | β dτ ≥ Z Y r | e − nR ( z ) p ( z ) | β dτ and (7.3) follows from the proof of Lemma 3.11. (cid:3) Let Y be a closed, unbounded subset of C with Y r regular for r sufficiently large. Let R be a superlogarithmic real-valued function on Y. We consider β ensembles P rob n,β,Q on Y . Theorem 7.3. Let Y and R be as above. Assume that the measure τ satisfies Hypotheses 3.9 and 7.1 and supp( τ ) = Y. Also assume thatHypothesis 6.1 holds. Then the sequence of probability measures definedby, for W an open subset of Y , ψ n ( W ) = P rob n,β,Q { z ∈ W } satisfyan l.d.p. with speed n and rate function J Y,β,Q ( z ) = β ( R ( z ) − V Y,R ( z )) = 2 Q ( z ) − V Y,R ( z ) . Proof. As noted in the proof of Theorem 6.2, J Y,β,Q ( z ) = 2 Q ( z ) − β ( Z Y log | z − t | dµ Y,β,Q ( t ) + ρ ) . To prove the l.d.p. will show that equation (6.1) holds, together withexponential tightness.We consider an open set W ⊂ Y. We may assume that W ⊂ Y r . Applying the results of section 6 to β ensembles on the compact set Y r for r sufficiently large, we have(7.8) inf { W | z ∈ W } lim n n log P rob n,β,Q { z ∈ W } = β ( V Y,R ( z ) − R ( z )) . We will show the same result holds for β ensembles on Y. Let N ⊂ Y be a compact neighbourhood of S ∗ R . Now A n,β,Q ( z ) and R W A n,β,Q ( z ) dτ ( z ), are, in each variable of the form e − nQ | p | β for apolynomial p ∈ P n . Hence by repeated use of Theorem 7.2 we have,( see also section 6)(7.9) Z Y n A n,β,Q ( z ) dτ ( z ) ≤ (1 + O ( e − cn )) Z N n A n,β,Q ( z ) dτ ( z ) , and(7.10) Z W Z Y n − A n,β,Q ( z ) dτ ( z ) ≤ (1 + O ( e − cn )) Z W Z N n − A n,β,Q ( z ) dτ ( z ) . Note that (7.9) shows that the integrals defining the normalizing con-stants (1.3) are finite. It also follows that taking N = Y r whether weconsider β ensembles on Y or Y r that lim n n log P rob n,β,Q { z ∈ W } willbe the same.To complete the large deviation property in the unbounded case wemust establish exponential tightness. That is :(7.11) lim n n log P rob n,β,Q {| z | > r } → −∞ as r → ∞ Proceeding as in the proof of Theorem 7.2 but without normalizingthe polynomial, we obtain(7.12) Z Y \ Y r | e − nR ( z ) p ( z ) | β dτ ≤ || e − nR ( z ) p ( z ) || βS R Z Y \ Y r dτ | z | nbβ ≤ C || e − nR ( z ) p ( z ) || βS R r − nbβ + a ARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 25 where C is independent of n, p. By the weighted BM inequality, we have(7.13) || e − nR ( z ) p ( z ) || βS R ≤ C (1 + ǫ ) n Z Y r | e − nR p | β dτ ≤ C (1 + ǫ ) n Z Y | e − nR p | β dτ. Now P rob n,β,Q {| z | > r } = 1 Z nβ,Q Z Y \ Y r Z Y n − A n,β,Q ( z ) dτ ( z )and using (7.12) and (7.13) we have P rob n,β,Q {| z | > r } ≤ C (1 + ǫ ) n r − nbβ + a Z nβ,Q Z Y n A n,β,Q ( z ) dτ ( z ) . But Z n,β,Q R Y n A n,β,Q ( z ) dτ ( z ) = 1 and since ǫ > (cid:3) Remark 7.4. Consider the probability distributions on Y, given by,for W an open subset of Y : ψ ′ n ( W ) := P rob n,β,Q { at least one of the coordinates z , z , . . . , z n ∈ W } . Then the sequence of probability distributions ψ ′ n under the hypothesisof Theorem 7.3 satisfy the same l.d.p. as ψ n . This is because the jointprobability distribution is symmetric in z , . . . , z n so nP rob { z ∈ W } ≥ P rob { at least one of z , . . . , z n ∈ W } ≥ P rob { z ∈ W } , and the l.d.p for ψ n has speed n. References [1] G. Anderson, A. Guionnet and O. 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