On equilibrium problems on the real axis. Applications Ram on - PowerPoint PPT Presentation

On equilibrium problems on the real axis. Applications Ram´ on Orive Universidad de La Laguna. The Canary Islands, Spain. e-mail: rorive@ull.es Based on joint works with A. Mart´ ınez–Finkelshtein (Univ. Almer´ ıa, Spain), E. A. Rakhmanov (Univ. South Florida, Tampa, USA) and Joaqu´ ın S´ anchez Lara (Univ. Granada, Spain). Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Motivation Many applications of Equilibrium Problems in the External fields: Asymptotics of orthogonal polynomials With respect to exponential weights With respect to general varying weights (connection with multipoint rational approximation) Asymptotics of Heine-Stieltjes polynomials Limit mean density of eigenvalues of Random Matrices Continuum limit of Toda lattice and Soliton Theory (KdV) Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Equilibrium Problem on a compact set K : compact subset of R . t > 0 , M t ( K ) : measures σ supported on K such that σ ( K ) = t . Under quite mild conditions on K , there exists a unique measure (Equilibrium or Robin measure) µ eq = µ eq,t , supp µ eq ⊂ K , minimizing the Energy � � I ( σ ) = − log | x − z | dσ ( x ) dσ ( z ) , σ ∈ M t ( K ) . supp µ eq = K . � V µ eq ( z ) = − log | x − z | dµ eq ( x ) = c t = const, z ∈ K . min z ∈ K V µ eq ( z ) = max σ ∈ M t ( K ) (min z ∈ supp σ V ( σ ; z )) . Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Equilibrium Problem on a compact set The simplest example: K = [ a, b ] ⊂ R , t = 1 Equilibrium measure dµ eq ( x ) = 1 dx � , x ∈ ( a, b ) π ( x − a )( b − x ) � � 4 V µ eq ( z ) = log , x ∈ ( a, b ) b − a Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Equilibrium Problem on a compact set A bit more involved example: K = [ a, b ] ∪ [ c, d ] ⊂ R , b < c , t = 1 Equilibrium measure dµ ( x ) = 1 ( x − ξ ) dx � , x ∈ ( a, b ) ∪ ( c, d ) π | ( x − a )( x − b )( x − c )( x − d ) | � c ( x − ξ ) dx ξ ∈ ( b, c ) , � = 0 ( x − a )( x − b )( x − c )( x − d ) b Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Equilibrium Problems in the presence of external fields Σ a closed subset of C (possibly unbounded) ϕ an “admissible” external field (Saff-Totik, 1997), ω ( z ) = e − ϕ ( z ) (weight). In particular, for an unbounded Σ , and certain t > 0 , it means: | z |→∞ , z ∈ Σ ( ϕ ( z ) − t log | z | ) = + ∞ . lim Then, there exists a unique measure µ ϕ = µ ϕ,t ∈ M t (Σ) minimizing the Weighted Energy � � I ϕ ( σ ) = − log ( | x − z | ω ( x ) ω ( z )) dσ ( x ) dσ ( z ) � � � = − log | x − z | dσ ( x ) dσ ( z ) + 2 ϕ ( x ) dσ ( x ) Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Equilibrium Problems in the presence of external fields S ϕ = S ϕ,t = = supp µ ϕ is a compact subset of Σ . Total (“chemical”) potential � = F ω = F ω,t , z ∈ S ϕ W µ ϕ ( z ) = V µ ϕ ( z ) + ϕ ( z ) ≥ F ω , z ∈ Σ S ϕ maximizes the F -functional (Mhaskar-Saff, Saff-Totik) � F ( K ) = t log cap ( K ) − ϕ ( x ) dµ eq,K ( x ) among all the compact subsets K of Σ . Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Equilibrium Problems in the presence of external fields Suppose that Σ ⊂ R . Then: ϕ convex = ⇒ S ϕ is an interval. ϕ real analytic = ⇒ S ϕ is comprised by a finite union of intervals. But... in general, finding the support S ϕ is a difficult task! Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Simple examples ϕ ( x ) = x 2 , t = 1 . ϕ is convex and symmetric = ⇒ S ϕ maximize F ( K ) , with K = [ − a, a ] , a > 0 = ⇒ S ϕ = [ − 1 , 1] and � ϕ ( x ) = 2 µ ′ 1 − x 2 π 3 x 4 , t = 1 . = ϕ ( x ) = 2 ⇒ S ϕ = [ − 1 , 1] and � 4 µ ′ 3 π (1 + 2 x 2 ) 1 − x 2 ϕ ( x ) = Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational External Fields q � ϕ ( x ) = P ( x ) + α j log | x − z j | , q ≥ 1 , z j ∈ C \ R , α j ∈ R , j =1 2 p − 1 � P ( x ) = x 2 p c j x j , p > 0 or P ≡ 0 , 2 p + j =1 q � α j ( x − Re z j ) ( x − z j )( x − z j ) = E ( x ) ϕ ′ ( x ) = P ′ ( x ) + D ( x ) , j =1 q � D ( x ) = ( x − z j )( x − z j ) . j =1 Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational External Fields If p > 0 ( ϕ has a polynomial part) = ⇒ ϕ is admissible for any t > 0 If p = 0 ( ϕ is “purely rational”) = ⇒ ϕ is weaker = ⇒ ϕ is admissible only for t ∈ (0 , T ) , q � T = α j j =1 Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational External Fields Equilibrium measure Size of the measure, “time” or “temperature” M t ( R ) = { σ : σ ( R ) = t > 0 } Support of the equilibrium measure k � S t = S ϕ,t = supp µ t = [ a 2 j − 1 , a 2 j ] , 1 ≤ k ≤ p + q j =1 Density of the equilibrium measure � 2 k � t ( z ) = 1 B ( z ) A ( z ) µ ′ , A ( z ) = ( z − a j ) π D ( z ) j =1 Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational External Fields First tool: Algebraic equation for the Cauchy Transform µ t ( z ) + ϕ ′ ( z )) 2 = R ( z ) = B ( z ) 2 A ( z ) (( − � , z ∈ C \ S t , D ( z ) 2 � dµ t ( y ) µ t ( z ) = � z − y 2( p + q ) − k − 1 2 k � � A ( z ) = ( z − a j ) , B ( z ) = ( z − b j ) , j =1 j =1 a 1 , . . . , a 2 k ∈ R , 1 ≤ k ≤ p + q. q � D ( x ) = ( x − z j )( x − z j ) j =1 Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational external fields Second tool: A dynamical viewpoint (Buyarov-Rakhmanov, 1999) Let t ∈ (0 , + ∞ ) . Except for a few values of t , µ t and its support S t depend analytically on t dµ t dt | t = t 0 = ω t 0 , ω t 0 : Robin measure (equilibrium measure in absence of external field) of the compact set S t 0 . = ⇒ A dynamical system for zeros of A and B : endpoints of S t and other zeros of the density!!! Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational External Fields Dynamical system for zeros of A and B a j = ∂a j 2 D ( a j ) F ( a j ) ˙ ∂t = − � k � = j ( a j − a k ) B ( a j ) , j = 1 , . . . , 2 k , b j = ∂b j D ( b j ) F ( b j ) ˙ ∂t = − � k � = j ( b j − b k ) A ( b j ) , j = 1 , . . . , 2( p + q ) − k − 1 , � a 2 j +1 F ( x ) F ∈ P k − 1 : � , dx = 0 , j = 1 , . . . , k − 1 . A ( x ) a 2 j q � D ( x ) = ( x − z j )( x − z j ) j =1 Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational external fields Singularities: collisions/bifurcations of zeros of A and/or B Singularity of type I: at a time t = T a real zero b of B (a double zero of R t ) splits into two simple zeros a − < a + , and the interval [ a − , a + ] becomes part of S t ( birth of a cut ). Phase transition: the number of cuts increases. Singularity of type II: at a time t = T two simple zeros a 2 s and a 2 s +1 of A (simple zeros of R t ) collide ( fusion of two cuts or closing of a gap ). Phase transition: the number of cuts decreases. Singularity of type III: at a time t = T a pair of complex conjugate zeros b and b of B (double zeros of R t ) collide with a simple zero a of A , so that λ ′ T ( x ) = O ( | x − a | 5 / 2 ) as x → a . No phase transition occurs: the number of cuts remains unchanged. Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Particular case: Polynomial external fields General Polynomial External Field 2 p − 1 � ϕ ( x ) = x 2 p t j x j , 2 p + t j ∈ R , j =1 Bleher, Eynard, Its, Kuijlaars, McLaughlin... Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Polynomial external fields A. Mart´ ınez Finkelshtein, RO, E. A. Rakhmanov (CMP, 2015) A suitably combined use of two ingredients = ⇒ Full description of dynamics in the Quartic case: ϕ ( x ) = x 4 4 + t 3 x 3 + t 2 x 2 + t 1 x . In particular: Two-cut is possible iff ϕ is a “sufficiently non-convex” external field: Simple geometrical characterization in terms of the relative position of critical points of ϕ Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

Rational External Fields with a polynomial part Motivation: Random Matrix Models G. S. Krishnaswami (2006): 1 -matrix model whose action is given by � � M 4 − log( v + M 2 ) V ( M ) = tr , Computable toy-model for the gluon correlations in a baryon background. Generalized Gauss-Penner model: ϕ ( x ) = ax 4 + bx 2 − c ln | x | , extending the classical Gauss-Pener model: ϕ ( x ) = x 2 − k ln | x | . = ⇒ ϕ ( x ) = x 4 − log( x 2 + v ) , v > 0 . Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.

A particular case: Generalized Gauss-Penner model RO, J. S´ anchez Lara (JMAA, 2015) ϕ ( x ) = αx 4 + βx 2 + γ log( x 2 + v ) , β, γ ∈ R , α, v > 0 , ⇓ φ ( x ) = 2 ϕ ( √ x ) = 2 αx 2 + 2 βx + 2 γ log( x + v ) , x ∈ [0 , + ∞ ) . ⇓ Simplified model in [0 , + ∞ ) φ ( x ) = 1 2 x 2 + βx + γ log( x + 1) , x ∈ [0 , + ∞ ) . Ram´ on Orive. rorive@ull.es MWAA, Bloomington, October 2015.