aa r X i v : . [ m a t h . C V ] O c t VOICULESCU’S ENTROPY AND POTENTIAL THEORY byThomas Bloom*Introduction
That the (negative of) the logarithmic energy of a planar measure can be ob-tained as a limit of volumes originated with work of D. Voiculescu ([Vo1], [Vo2]).His motivation came from operator theory and free probability theory. Ben Arousand A. Guionnet [Be-Gu] put that result in the framework of large deviations.Other results in that direction are due to Ben Arous and Zeitouni [Be-Ze] and Hiaiand Petz [Hi-Pe]. These authors use potential theory and retain the basic form ofVoiculescu’s original proof.Informally, these results express the asymptotic value (as d → ∞ ) of the aver-age of a “weighted” VanDerMonde determinant of a point ( λ , · · · , λ d ) ∈ E d , asthe discrete measures κ d ( λ ) := 1 d d X j =1 δ ( λ j ) approach a fixed probability measure µ with compact support E in C . Such weighted VanDerMonde (VDM) determi-nants arise, for example, as the joint probability distribution of the eigenvalues ofcertain ensembles of random Hermitian matrices and also in the study of certaindeterminental point processes. Specifically, we prove, for measures µ with supportin a rectangle H : * Partially supported by an NSERC of Canada Grant. Typeset by AMS -TEX Theorem 3.1. inf G ∋ µ lim d →∞ d log Z ˜ G d ( µ ) | VDM wd ( λ ) | dτ ( λ ) = Σ( µ ) − Z Qdµ where the infimum is over all neighborhoods of µ in the weak* topology, ˜ G d ( µ ) := { λ ∈ H d | κ d ( λ ) ∈ G } , Q = − log w, Σ( µ ) = Z Z log | z − t | dµ ( z ) dµ ( t ) , τ is ameasure satisfying an appropriate density condition on H (prop 3.1), and VDM wd ( λ ) is a weighted VanDerMonde determinant (see (2.7)) with w continuous and > on H . This result is not essentially new however the proof is new. The lower bound intheorem 3.1 is obtained by using Markov’s polynomial inequality on the weightedVanDerMondes when the weight is a real polynomial, the general case being ob-tained by approximation.Voiculescu’s method (and those of the authors cited above) uses a “discretiza-tion” argument on the measure µ (the method has been used in other situations([Ze]-[Ze])). This method relies on the factorization of the VDM determinent intolinear factors. The method of this paper does not use such factorization-the interestin doing so, being in higher dimensional versions of these results (The methods ofthis paper were the basis for the announcement of some higher dimensional results[Bl, talk]).R. Berman ([Be1], [Be2]) has recently proven large deviation results and a ver-sion of the above result in general higher dimensional situations. Reduced to theone-dimensional case of compact subsets of C , his proof is different than that ofVoiculescu or this paper. Acknowledgement
I would like to thank R. Berman, L. Bos and N. Levenberg for helpful discussionsconcerning this paper.
OICULESCU’S ENTROPY AND POTENTIAL THEORY 3
1. Topology on M ( E )Let E be a closed subset of C (which we identify with R ). We let M ( E ) denotethe set of positive Borel probability measures on E with the weak* topology.The weak* topology on M ( E ) is given as follows (see [E], appendix A8). Aneighborhood basis of any µ ∈ M ( E ) is given by sets of the form(1.1) { ν ∈ M ( E ) (cid:12)(cid:12)(cid:12) | Z E f i ( dµ − dν ) | ≤ ǫ for i = 1 , · · · , k } where ǫ > f , · · · , f k are bounded continuous functions on E . M ( E ) is a complete metrizable space and for E compact a neighborhood basisof µ ∈ M ( E ) is given by sets of the form(1.2) G ( µ, k, ǫ ) := { ν ∈ M ( E ) (cid:12)(cid:12)(cid:12) | Z E x n y n ( dµ − dν ) | < ǫ } for k, n , n ∈ N , n + n ≤ k and ǫ > . That is G ( µ, k, ǫ ) consists of all probability measures on E whose (real) moments,up to order k , are within ǫ of the corresponding moment for µ .It is clear that for k ≥ k and ǫ ≤ ǫ that(1.3) G ( µ, k , ǫ ) ⊂ G ( µ, k, ǫ ) . Now for λ = ( λ , · · · , λ d ) ∈ C d , we let(1.4) κ d ( λ ) := 1 d d X j =1 δ ( λ j )where δ is the Dirac δ -measure at the indicated point.We let(1.5) ˜ G d ( µ, k, ǫ ) := { λ ∈ E d (cid:12)(cid:12)(cid:12) κ d ( λ ) ∈ G ( µ, k, ǫ ) } . It follows from (1.3) that(1.6) ˜ G d ( µ, k , ǫ ) ⊂ ˜ G d ( µ, k, ǫ ) for k ≥ k and ǫ ≤ ǫ. For λ ∈ C d we let(1.7) ∆ d ( λ ) = { λ ′ ∈ C d (cid:12)(cid:12)(cid:12) | λ ′ j − λ j | ≤ e −√ d for j = i, · · · , d } . Proposition 1.1 and 1.2 follow immediately from the definition of the weak*topology in M ( E ) (for E compact). Proposition 1.1.
Let f be continuous on E and µ ∈ M ( E ) . Given ǫ > thereexist k, ǫ such that (cid:12)(cid:12)(cid:12) Z E f ( dµ − κ d ( λ )) (cid:12)(cid:12)(cid:12) ≤ ǫ for λ ∈ ˜ G d ( µ, k, ǫ ) . Proposition 1.2.
Let ν ∈ G ( µ, k, ǫ ) . Then there exists k , ǫ , such that G ( ν, k , ǫ ) ⊂ G ( µ, k, ǫ ) . Proposition 1.3.
Let λ ∈ ˜ G d ( µ, k, ǫ ) . Then ∆ d ( λ ) ∈ ˜ G d ( µ, k, ǫ ) for all d suffi-ciently large. Proof:
The proof follows from the fact that monomials satisfy a Lipshitz conditionon E .
2. Markov’s Polynomial Inequality
The classical Markov polynomial inequality for real polynomials on an interval I ⊂ R is an estimate for the derivative of the polynomial in terms of its degree andsup norm on I . Specifically ([Be-Er], theorem 5.1.8)(2.1) | p ′ ( x ) | ≤ Ak k p k I for x ∈ I where k = deg( p ) and A is a constant >
0. For I = [ − ,
1] on may take A = 1.Numerous extensions of (2.1) to multivariable settings have been established (seee.g. [Ba], [Pl]). OICULESCU’S ENTROPY AND POTENTIAL THEORY 5
We will however use a version of (2.1) for rectangles H ⊂ R which is an im-mediate consequence of (2.1). (We will always assume that rectangles have sidesparallel to the axes). Let p ( x, y ) be a polynomial of degree ≤ k in each varaible,then(2.2) | grad ( p )( x ) | ≤ Ak k p k H where A > z to z in H we have(2.3) | p ( z ) − p ( z ) | ≤ Ak k p k H | z − z | . We will now use (2.3) to show in quantitative terms that the value of polynomialsat points near a point where it assumes its maximum is close to the maximum value.Let { Λ d } d =1 , ··· be a sequence of polynomials on ( R ) d , non negative on H d , suchthat for some constants c > , γ > d is of degree ≤ c d γ ineach of its 2 d real variables. Let z M := ( z M , · · · , z Md ) be a point in H d ⊂ C d ≃ R d where Λ d assumes its maximum i.e. Λ d ( z M ) = k Λ d k H d . Theorem 2.1.
For z ∈ ∆ d ( z M ) ∩ H d . Then Λ d ( z ) ≥ Λ d ( z M ) ψ ( d ) where ψ ( d ) = 1 − cd γ e −√ d for some constants c, γ > (independent of d ). Proof:
We write Λ d ( z M ) − Λ d ( z ) in the form(2.4)Λ d ( z M ) − Λ d ( z ) = d X j =1 Λ d ( z , · · · , z j − , z Mj , · · · , z Md ) − Λ d ( z , · · · , z j , z Mj +1 , · · · , z Md ) . But for z , · · · , z j − , z Mj +1 , · · · , z Md fixed, t → Λ d ( z , · · · , z j − , t, z Mj +1 , · · · , z Md ) is apolynomial in R of deg ≤ c d γ in each real variable. Applying (2.3) and the fact that z ∈ ∆ d ( z M ) to each term on the right side of (2.4) we have an estimate of theform(2.5) Λ d ( z M ) − Λ d ( z ) ≤ dA ( c d γ ) Λ d ( z M ) e −√ d . The result follows. (cid:3)
We will apply this result to sequences of polynomials constructed as follows: Let(2.6) VDM d ( λ ) = VDM d ( λ , · · · , λ d ) = Y ≤ i 3. Energy as a limit of volumes Let E be a compact subset of C and w an admissible weight function on E (i.e. w is uppersemicontinuous, w ≥ , w > E . In particular, E is non-polar.The weighted equilibrium measure (see [Sa-To], theorem I 1.3), denoted µ eq ( E, w )is the unique probability measure which minumizes the functional I w ( ν ) over all ν ∈ M ( E ) where(3.1) I w ( ν ) : = Z Z log (cid:16) | z − t | w ( z ) w ( t ) (cid:17) dν ( z ) dν ( t )= − Z Z log | z − t | dν ( z ) dν ( t ) + 2 Z Q ( z ) dν ( z ) OICULESCU’S ENTROPY AND POTENTIAL THEORY 7 where(3.2) Q ( z ) := − log w ( z ) .I w ( ν ) is termed the weighted energy of the measure ν . We also use the notation(3.3) Σ( ν ) := Z Z log | z − t | dν ( z ) dν ( t ) . Σ( ν ) is termed the free entropy of ν (it may assume the value −∞ ). We let(3.4) δ wd := Max λ ∈ E d | VDM wd ( λ ) | d ( d − . Then (see [Sa-To], chapter III, theorem 1.1)(3.5) δ w := lim d →∞ δ wd exists and(3.6) log δ w = − I w ( µ eq ( E, w )) = Σ( µ eq ( E, w )) − Z Q ( z ) dµ eq ( E, w ) . Now, let τ be a positive Borel measure on E .We say that the triple ( E, w, τ ) satisfies the weighted Bernstein-Markov (B-M)inequality if, for all ǫ > 0, these exists a constant c > p of degree ≤ k we have(3.7) k w k p k E ≤ c (1 + ǫ ) k k w k p k L ( τ ) . We set(3.8) Z d := Z E d | VDM wd ( λ ) | dτ ( λ ) . where dτ ( λ ) = dτ ( λ ) · · · dτ ( λ d ) is the product measure on E d . Then if ( E, w, τ )satisfies the weighted B-M inequality ([Bl-Le2]).(3.9) lim d →∞ Z d − d = δ w . We will need measures on E which satisfy the weighted B-M ineqality for allcontinuous admissible weights.To this end we consider measures τ which satisfy the following condition (satisfiedby any measure that is a positive continuous function times Lebesgue measure):There is a constant T > τ ( D ( z , r )) ≥ r T for all z ∈ E and r ≤ r . Here D ( z , r ) denotes the disc center z radius r and r > Proposition 3.1. Let H be a rectangle in C and let τ satisfy (3.10). Then for allcontinuous functions w > on H, ( H, w, τ ) satisfies the weighted B-M inequality. Proof: First we can consider H ⊂ C ≃ R as a subset of C . Then, using[Bl-Le1], theorem 2.2 and [B1], theorem 3.2. ( H, w, τ ) satisfies the weighted B-Minequality as a subset of C (the definition of which is an obvious adaptation of(3.7) to the several variable case-see [Bl]). But every analytic polynomial p ( z ) on C is the restriction to C ≃ R ⊂ C of the analytic polynomial p ( z + iz ). Hencethe result (cid:3) Let H be a rectangle in C , µ ∈ M ( H ), τ satisfy (3.10), and let φ > H . Let S = − log φ . We will consider integrals of the form(3.11) J φd ( µ, k, ǫ ) := Z ˜ G d ( µ,k,ǫ ) | VDM φd ( λ ) | dτ ( λ ) . The integral in (3.11) is of the same form as that in (3.8) used to define Z d however here we only integrate over a subset of H d . Theorem 3.1 below establishesasymptotic properties of such integrals. The leading term depends only on µ on S (and as mentioned in te introduction, the result is not essentially new but goesback to results of Voiculescu ([Vo1], [Vo2]). OICULESCU’S ENTROPY AND POTENTIAL THEORY 9 Theorem 3.1. inf k,ǫ n lim d →∞ d log J φd ( µ, k, ǫ ) o = Σ( µ ) − Z Sdµ Proof: To prove this result we will show(a) inf k,ǫ n lim d →∞ d log J φd ( µ, k, ǫ ) o ≤ Σ( µ ) − Z Sdµ and(b) inf k,ǫ n lim d →∞ d log J φd ( µ, k, ǫ ) o ≥ Σ( µ ) − Z Sdµ. To prove the upper bound (a) we will follow ([Be1], proposition 3.4). The proofdoes not use (3.10). Let w be continuous > H . Then(3.12) d Q i =1 w ( λ i ) d | VDM φd ( λ ) | = | VDM wd ( λ ) | d Q i =1 φ ( λ i ) d . Hence,(3.13) | VDM φd ( λ ) | ≤ ( δ wd ) d ( d − exp (cid:0) d Z H ( Q − S ) κ d ( λ ) (cid:1) . Let λ d ∈ ˜ G d ( µ, k, ǫ ) be a point at which the maximum of | VDM φd ( λ ) | over˜ G d ( µ, k, ǫ ) is attained. (3.13) implies that(3.14) J φd ( µ, k, ǫ ) τ ( H ) d ≤ ( δ wd ) d ( d − exp(2 d Z H ( Q − S ) κ d ( λ d )) . For any sequence of d ’s we may pass to a subsequence and assume that thesequence of measures κ d ( λ d ) converges to a measure σ ∈ G ( µ, k, ǫ ).We deduce that(3.15) lim d →∞ d log J φd ( µ, k, ǫ ) ≤ log δ w + 2 Z H ( Q − S ) dσ. Taking the inf over k, ǫ , the σ ’s converge to µ soinf k,ǫ lim 1 d log J φd ( µ, k, ǫ ) ≤ log δ w + 2 Z H ( Q − S ) dµ. Now take a sequence of continuous weights w such that µ eq ( H, w ) converges to µ in M ( H ) and Σ( µ eq ( H, w )) converges to Σ( µ ) (see proof of (b) (iii)).Then using (3.6) we obtain (a).For the lower bound (b) we proceed as follows.We prove (b) when(i) µ = µ eq ( H, w ), w is a polynomial > φ = w .(ii) µ as in (i) but the restriction on φ is dropped.(iii) general µ .(i) We consider points z M ∈ H d at which | VDM wd ( λ ) | assumes its maximum (i.e. w -Fekete points). It is known that κ d ( z M ) converges to µ in M ( H ) so, for d large, κ d ( z M ) ∈ ˜ G d ( µ, k, ǫ ). Then for d sufficiently large, using proposition 1.3(3.16) J wd ( µ, k, ǫ ) ≥ τ (∆ d ( z M ))Min λ ∈ ∆ d ( z M ) ∩ H | VDM wd ( λ ) | . By (3.10) τ (∆ d ( z M )) ≥ e − T d √ d and using theorem 2.1 on the sequence of poly-nomials | VDM wd ( λ ) | we have(3.17) lim d →∞ d log J wd ( µ, k, ǫ ) ≥ log δ w = Σ( µ ) − Z Qdµ. (ii) Given ǫ > 0, by proposition 1.1, choose k, ǫ so that(3.18) Z ( Q − S )( dµ − κ d ( λ )) ≤ ǫ, for all λ ∈ ˜ G d ( µ, k, ǫ ) . This yields(3.19) d Q i =1 w ( λ i ) d ≤ d Q i =1 φ ( λ i ) d exp(2 d [ ǫ − Z ( Q − S ) dµ ]) . Multiplying by | VDM d ( λ ) | and integrating over ˜ G d ( µ, k, ǫ ) gives(3.20) exp(2 d [ − ǫ + Z ( Q − S ) dµ ]) J wd ( µ, k, ǫ ) ≤ J φd ( µ, k, ǫ ) . OICULESCU’S ENTROPY AND POTENTIAL THEORY 11 Then using (i) and the fact that ǫ > k,ǫ lim d →∞ n d log J φd ( µ, k, ǫ ) o ≥ log δ w + 2 Z ( Q − S ) dµ and using (3.6) completes (ii).For (iii) we will use an approximation argument. First we note that it is animmediate consequence of proposition 1.2 that µ → inf k,ǫ lim d →∞ d log J φd ( µ, k, ǫ ) isuppersemicontinuous on M ( H ). So, it suffices to show that any µ ∈ M ( H ) maybe approximated by measures { µ s } ∈ M ( H ) where each µ s satisfies (i) above andΣ( µ s ) converges to Σ( µ ). ( µ → Σ( µ ) is uppersemicontinuous on M ( H ) but not,in general, continuous). First, we may assume supp( µ ) ⊂ int( H ) since, taking H centered at 0 the measures µ s = π ∗ s ( µ ), the push forward of µ under the scaling z → sz ( s < 1) satisfy Σ( µ s ) converges to Σ( µ ). Next for µ with compact support inint( H ) we approximate µ by µ s = µ ∗ ρ s where ρ = ρ ( | z | ) is a standard smoothingkernel for subhamonic functions on C and ρ s = s − ρ (cid:16) | z | s (cid:17) . Then ρ ( | z | ) ρ ( | t | ) isa standard smoothing kernel for plurisubhamonic functions on C so log | z − t | ∗ (cid:16) ρ z ( | z | ) ρ s ( | t | ) (cid:17) decreases pointwise to log | z − t | .Now, for ν a positive measure with compact support in R n , ψ a smooth functionwith compact support such that ψ ( x ) = ψ ( − x ) and h ∈ L ( R n ) then Z R n ψ ( ν ∗ h ) dm = Z R n ( ψ ∗ h ) dν where dm denotes Lebesque measure and ∗ convolution.Applying this formula to R ≃ C with ( z, t ) as coordinates, ν = µ ⊗ ν , ψ = ρ s ( | z | ) ρ z ( | t | ) and h = log | z − t | , then using the Lebesgue monotone convergencetheorem yields Σ( µ s ) → Σ( µ ) (as s → µ a smooth function with compact support times Lebesgue measure let Q be a smooth potential for µ which,adding a constant, we may assume is < H . Then µ = µ eq ( H, w ) where w = e − Q and one may approximate µ by µ s = µ eq ( H, w s ) where w s are realpolynomial weights converging uniformly to w on H . To see that Σ( µ s ) convergesto Σ( µ ) we may use ([Sa-To], theorem 6.2 (c), chapter I) - which is stated formonotonically decreasing sequences of weights but the conclusion also holds foruniformly convergent sequences of weights. 4. Entropy Let µ ∈ M ( H ). The free entropy of µ (see (3.3)) defined as an integral may beobtained via discrete measures as follows:Let W ( µ ) be defined via(4.1) W d ( µ, k, ǫ ) := sup {| VDM d ( λ ) | d ( d − (cid:12)(cid:12)(cid:12) κ d ( λ ) ∈ ˜ G d ( µ, k, ǫ ) } and let(4.2) W ( µ, k, ǫ ) = lim d →∞ W d ( µ, k, ǫ )and(4.3) W ( µ ) = inf k,ǫ W ( µ, k, ǫ ) . Then Theorem 4.1. log W ( µ ) = Σ( µ ) . Proof: The proof consists of establishing the two inequalities(a) log W ( ν ) ≤ Σ( µ ) and(b) Σ( µ ) ≤ log W ( µ ) . OICULESCU’S ENTROPY AND POTENTIAL THEORY 13 For (a) let κ d ( λ d ) = d P dj =1 δ ( λ dj ) be, for d = 1 , , · · · , a sequence of discretemeasures converging to µ weak* such that(4.4) log W ( µ ) = lim d →∞ d X j = k log | λ dj − λ dk | . Now,(4.5) lim d →∞ d X j = k δ ( λ dj , λ dk ) = µ ⊗ µ weak ∗ and so, since log | z − t | is u.s.c.(a) follows from ([Sa-To], theorem 1.4, chapter O).For (b) we note that by definition of the quatities involved J d ( µ, k, ǫ ) ≤ W d ( µ, k, ǫ ) d ( d − τ ( H ) d . so that lim d →∞ d log J d ( µ, k, ǫ ) ≤ log W ( µ, k, ǫ )Taking the inf over k, ǫ and using theorem 3.1, (b) follows. (cid:3) Corollary 4.1. Define W φd ( µ, k, ǫ ) analogously to the definition of W d ( µ, k, ǫ ) in(4.1). That is W φd ( µ, k, ǫ ) = sup {| VDM φd ( λ ) | d ( d − | κ d ( λ ) ∈ ˜ G d ( µ, k, ǫ ) } and define W φ ( µ ) analogously to the definition of W ( µ ) (see (4.3)). Then Σ( µ ) − Z Sdµ = log W φ ( µ ) . 5. Large Deviation Consider the sequence of probability measures on H d (for d = 1 , , · · · ) given by(5.1) | VDM φd ( λ ) | dτ ( λ ) Z φd := Prob d . Then(5.2) 1 d log Prob d ( ˜ G d ( µ, k, ǫ )) = Prob d { λ | κ d ( λ ) ∈ G ( µ, k, ǫ ) } . Using theorem 3.1 and (3.9) givesinf k,ǫ lim d →∞ d log Prob d ( ˜ G d ( µ, k, ǫ )) = I φ ( µ eq ( H, φ )) − I φ ( µ ) . The functional µ → I φ ( µ ) − I φ ( µ eq ( H, φ )) =: I ( µ ) attains its minimum value ofzero at the unique measure µ = µ eq ( H, φ ).Then I ( µ ) is a good rate functional and the sequence of descrete random mea-sures κ d ( λ ) satisfy a large devation principle in the scale d − (see discussion [Hi-Pe],page 211). OICULESCU’S ENTROPY AND POTENTIAL THEORY 15 References [Ba] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact setsin R n , Proc Am. Math. Soc. (1995), 485-494.[Be1] R. Berman, Large deviations and entropy for determinental point processes on complexmanifolds. arXiv:0812.4224 .[Be2] R. Berman, Determinental point processes and fermions on complex manifolds: bulkuniversity arXiv:0811.3341 .[B1-Le1] T. Bloom and N. Levenberg, Capacity convergence results and applications to a Bernstein-Markov inequality , Tr. Am. Math Soc. 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