Coulomb gas ensembles in 2D H. Hedenmalm December 11, 2015 H. Hedenmalm Coulomb gas ensembles in 2D
Coulomb gas in 2D We consider n repelling particles in 2D confined by a potential V : C → R . The interaction energy between the repelling particles is modelled by � 1 E int := log | z j − z k | , V j , k : j � = k where z j denotes the position of the j -th particle, and the potential energy is given by n � E pot := V ( z j ) . V j =1 The total energy of a configuration ( z 1 , . . . , z n ) ∈ C n is then given by V + E pot E V := E int V . H. Hedenmalm Coulomb gas ensembles in 2D
Coulomb gas. Gibbs model and inverse temperature In any reasonable gas dynamics model, the low energy states should be more likely than the high energy states. Fix a positive constant β , and let Z n be the constant (“partition function”) � C n e − β 2 E V dvol 2 n , Z n := where vol 2 n denotes standard volume measure in C n ∼ = R 2 n . Here, we need to assume that V grows at sufficiently at infinity to make the integral converge. The Gibbs model gives the joint density of states 1 e − β 2 E V , Z n which we use to define a probability point process Π n ∈ prob ( C n ) by setting d Π n := 1 e − β 2 E V dvol 2 n . Z n H. Hedenmalm Coulomb gas ensembles in 2D
Simulation of the Ginibre ensemble V ( z ) = m | z | 2 (1700 pts) 1.5 n=1700 1 0.5 0 -0.5 -1 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 H. Hedenmalm Coulomb gas ensembles in 2D
Electron cloud interpretation. Marginal measures The process Π n models a cloud of electrons in a confining potential. Clearly, Π n is random probabilty measure on C n . In order to study this process as n → + ∞ , it is advantageous to introduce the marginal probability measures Π ( k ) (for 0 ≤ k ≤ n ) given by n Π ( k ) n ( e ) := Π n ( e × C n − k ) , for Borel measurable subsets e ⊂ C k . In particular, Π ( n ) = Π n . n The associated measures n ! Γ ( k ) ( n − k )!Π ( k ) := n n are called intensity (or correlation) measures . To simplify the notation, we write Γ n := Γ ( n ) n . H. Hedenmalm Coulomb gas ensembles in 2D
Aggregation of quantum droplets. Monotonicity It is of interest to analyze what the addition of one more particle means for the process. THEOREM 1. If β = 2, then ∀ k : Γ ( k ) ≤ Γ ( k ) n +1 . n This means that for the special inverse temperature β = 2, the addition of a new particle monotonically increases all the intensities. REMARK 2. The assertion of Theorem 1 fails for β > 2. For β < 2, however, we conjecture that the assertion of Theorem 1 remains valid. H. Hedenmalm Coulomb gas ensembles in 2D
The determinantal nature of β = 2 case (1) The proof of Theorem 1 is based on the fact that the point process Π n is determinantal for β = 2. To explain what this means, we need the space Pol n of all polynomials in z of degree ≤ n − 1. We equip Pol n with the inner product structure of L 2 ( C , e − V ). Then under standard assumptions on V , point evaluations are bounded, and we obtain elements K w ∈ Pol n such that p ( w ) = � p , K w � L 2 ( C , e − V ) . The function K ( z , w ) := K w ( z ) may be written in the form n − 1 � K ( z , w ) = e j ( z )¯ e j ( w ) , j =0 where the e j form an ONB. It is called the reproducing kernel . H. Hedenmalm Coulomb gas ensembles in 2D
The determinantal nature of β = 2 case (2) The determinantal structure of the process is easiest to see by considering intensities: j V ( z j ) det[ K ( z i , z j )] k d Γ ( k ) n ( z ) = e − � i , j =1 . For instance, if we are interested in the intensity Γ (1) n , we should analyze K ( z , z ) e − V ( z ) . The expression u n ( z ) := 1 nK ( z , z ) e − V ( z ) is called the 1 -point function . The determinantal case β = 2 models Random Normal Matrices . H. Hedenmalm Coulomb gas ensembles in 2D
Renormalization of the potential To obtain a reasonable limit as n → + ∞ , we need to renormalize the potential. So we put V := mQ , where the parameter m is essentially proportional to n as n tends to infinity. Here, Q is a fixed confining potential. N. B. Note that in the determinantal case, we just need to analyze the (polynomial) reproducing kernels K ( z , w ) for the space of polynomials of degree ≤ n − 1 with respect to the weight e − mQ in the plane C . H. Hedenmalm Coulomb gas ensembles in 2D
Approximation of the energy (1) We recall that n � � 1 mQ + E pot E mQ = E int mQ = log | z j − z k | + m Q ( z j ) , j , k : j � = k j =1 so that n � � E mQ = 1 | z j − z k | + m 1 log Q ( z j ) . n 2 n 2 n 2 j , k : j � = k j =1 If n / m = τ , then n � � E mQ = 1 | z j − z k | + 1 1 log Q ( z j ) . n 2 n 2 n τ j , k : j � = k j =1 H. Hedenmalm Coulomb gas ensembles in 2D
Approximation of the energy (2) If we put (for probability measures σ ) � � � 1 I Q [ σ ] := log | ξ − η | d σ ( ξ ) d σ ( η ) + Q d σ, C C C then E mQ ≈ I Q /τ [ σ ] , n 2 where n � d σ = 1 d δ z j . n j =1 Here, “ ≈ ” means that we disregard the singularities which appear from diagonal terms in the integral. We write I ♯ Q /τ [ σ ] to indicate that we have removed the singular diagonal part from I Q /τ [ σ ]. H. Hedenmalm Coulomb gas ensembles in 2D
The Gibbs model and energy heuristics We recall the density of states from the Gibbs model d Π n := 1 2 E ( λ 1 ,...,λ n ) dvol 2 n = 1 2 I ♯ e − n 2 β e − β Q /τ [ σ ] dvol 2 n . Z n Z n The factor n 2 in the exponent means that high energy states get severely punished and we expend generally convergence to the lowest energy state. To make this more precise, let ˆ σ τ ∈ prob c ( C ) minimize min σ I Q /τ [ σ ] . The measure ˆ σ τ is called the equilibrium measure . H. Hedenmalm Coulomb gas ensembles in 2D
Johansson’s marginal measure theorem THEOREM 3 . Under minimal growth and smoothness assumptions on Q , we have for fixed k that Π ( k ) σ ⊗ k → ˆ as n → + ∞ , while n = m τ + o ( m ) , n τ in the weak-star sense of measures. REMARK 4 . In particular, the 1-point function converges to the equilibrium density. Theorem 3 was obtain by K. Johansson in the case of Coulomb gas on the real line [J1]. His techniques work also in the planar case, with some modifications [HM1]. H. Hedenmalm Coulomb gas ensembles in 2D
Obstacle problem and the equilibrium measure We consider the obstacle problem ˆ Q τ ( z ) := sup { q ( z ) : q ≤ Q on C , q ∈ Subh τ ( C ) } , where Subh τ ( C ) denotes the convex set of subharmonic functions u : C → [ −∞ , + ∞ [ with u ( z ) ≤ 2 τ log + | z | + O (1) . For a measure σ , its logarithmic potential U σ is � 1 U σ ( ξ ) := 2 log | ξ − η | d σ ( η ) . C THEOREM 5 (Frostman) For some constant c , σ τ . ˆ Q τ = c − τ U ˆ H. Hedenmalm Coulomb gas ensembles in 2D
The support of the equilibrium measure Let S τ := supp ˆ σ τ . This is called the (spectral) droplet . THEOREM 6 (Kinderlehrer-Stampacchia theory) Under smoothness on Q , we have ∆ ˆ Q τ = 1 S τ ∆ Q , so that σ τ = 1 S τ ∆ Q d ˆ . 4 πτ REMARK 7 It follows that the study of the dynamics of the equilibrium measures ˆ σ τ reduces to the study of the supports S τ . This is in contrast with the 1D theory. H. Hedenmalm Coulomb gas ensembles in 2D
Comparison with Hermitian ensebles If we consider the degenerate case when Q = + ∞ on C \ R , we get the usual Hermitian ensebles (the eigenvalues are forced to be real). This can be thought of as a limit of smooth potentials � Q ( x + i y ) := Q ( x ) + ay 2 , where we let a → + ∞ . We expect that the droplets S a tend to a compact subset of R as a → + ∞ , where the eigenvalues accumulate, and that the local vertical width of S a corresponds to the local density of eigenvalues in the Hermitian ensemble. The relation σ τ = 1 4 π ∆ ˆ τ d ˆ Q τ d A should survive also in the Hermitian case, although the right hand side must be understood in the sense of distribution theory. E.g., the Wigner semi-circle law comes from an obstacle problem with Q ( x ) = x 2 along the real line and Q = + ∞ elsewhere in C . H. Hedenmalm Coulomb gas ensembles in 2D
Linear statistics We now mention an application of Johansson’s marginal measure theorem (Theorem 3) to linear statistics. For f ∈ C b ( C ), put tr n f := f ( z 1 ) + · · · + f ( z n ) . THEOREM 8 Under the assumptions of Theorem 3, we have the convergence � 1 n tr n f → f d ˆ σ τ C in all moments as m → + ∞ and n = m τ + o ( m ). REMARK 9 We may interpret this as the statement that when applied to a test function, the empirical measure converges to the equilibrium measure. H. Hedenmalm Coulomb gas ensembles in 2D
Fluctuations (1) We now fix τ = 1 and write S = S 1 . In the context of Theorem 8, with smooth compactly test functions f , we would like to analyze first E tr n f − n � f , ˆ σ � . THEOREM 10 Under smoothness of Q and simple-connectedness of S , and smoothness of ∂ S , σ � → 1 8 π � f , ∆(1 S + L S ) � , E tr n f − n � f , ˆ where L := log ∆ Q , and L S is the harmonic extension to the outside of L | S . H. Hedenmalm Coulomb gas ensembles in 2D
Fluctuations (2) The next level to understand is fluctuations . THEOREM 11 Under smoothness of Q and simple-connectedness of S , and smoothness of ∂ S , tr n f − E tr n f → N (0 , s 2 ) , where � s 2 = 1 |∇ f S | 2 dvol 2 . 4 π C H. Hedenmalm Coulomb gas ensembles in 2D
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