Free energy of a particle in high-dimensional Gaussian potentials - PowerPoint PPT Presentation

Free energy of a particle in high-dimensional Gaussian potentials with isotropic increments Anton Klimovsky EURANDOM Eindhoven University of Technology September 1, 2011, Prague http://arxiv.org/abs/1108.5300

Gaussian fields with isotropic increments ◮ Gaussian random field : X N = { X N ( u ) : u ∈ ❘ N } . ◮ X N ( u ) centred Gaussian , u ∈ ❘ N . ◮ Isotropic increments : � 1 � ( X N ( u ) − X N ( v )) 2 � N � u − v � 2 = : D N ( � u − v � 2 u , v ∈ ❘ N . � = D 2 ) , ❊ 2 ◮ NB! N ≫ 1 . ◮ Any (admissible) D : ❘ + → ❘ + .

Complete classification of the correlators A.M. Yaglom (1957): 1. Isotropic field: � 1 � N � u − v � 2 ❊ [ X N ( u ) X N ( v )] = B , u , v ∈ Σ N , 2 � + ∞ − t 2 r � � B ( r ) = c 0 + ν ( d t ) , exp 0 c 0 ∈ ❘ + , ν ∈ M finite ( ❘ + ) . D ( r ) = 2 ( B ( 0 ) − B ( r )) . 2. Non-isotropic field with isotropic increments: � + ∞ − t 2 r � � �� D ( r ) = 1 − exp ν ( d t )+ A · r , r ∈ ❘ + , 0 A ∈ ❘ + , ν ∈ M (( 0; + ∞ )) � + ∞ t 2 ν ( d t ) t 2 + 1 < ∞ . 0

A particle subjected to a rugged potential: ◮ Particle state space : √ S N : = { u ∈ ❘ N : � u � 2 ≤ L S N : = S N , S ⊂ ❘ , N } , L > 0 . or ◮ Partition function : √ � � � Z N ( β ) : = µ N ( d u ) exp β NX N ( u ) , β ∈ ❘ + . S N ◮ Log-partition function : p N ( β ) : = 1 N log Z N ( β ) . ◮ Q: N → + ∞ p N ( β ) = : p ( β ) = ? lim

Parisi-type functional ◮ Regularised derivative: � D ′ ( r ) , r ∈ [ 1 / M ; + ∞ ) , D ′ , M ( r ) : = M , r ∈ [ 0;1 / M ) . ◮ Parisi terminal value problem: � qq f ( y , q )+ x ( q )( ∂ y f ( y , q )) 2 � � ∂ q f ( y , q )+ 1 2 D ′ , M ( 2 ( r − q )) ∂ 2 = 0 , f ( y , 1 ) = h ( y ) , q ∈ ( 0 , r ) , y ∈ ❘ . ◮ Spin glass order parameter: x ∈ X ( r ) : = { x : [ 0 , r ] → [ 0 , 1 ] | càdlàg ↑ , x ( 0 ) = 0 , x ( r ) = 1 } . ◮ Boundary conditions (product state space) � β uy + λ u 2 � � h λ ( y ) : = log S µ ( d u ) exp , y ∈ ❘ , λ ∈ ❘ . ◮ Parisi-type functional: � 1 − β 2 � � � f ( M ) � 0 x ( q ) d θ ( M ) P ( β , r )[ x ] : = lim r , x , h λ ( 0 , 0 ) − λ r inf ( q ) , r 2 M ↑ + ∞ λ ∈ ❘ θ ( M ) ( q ) : = − qD ′ , M ( q ) − D ( q ) , q ∈ ❘ + .

Variational formula Effective size of the state space: � � 1 � u � 2 d : = sup N sup . 2 u ∈ S N N Theorem almost surely and in L 1 . p ( β ) : = sup x ∈ X ( r ) P ( β , r )[ x ] , inf r ∈ [ 0; d ]

Heuristics: "localisation" ◮ Covariance structure: ❊ [ X N ( u ) X N ( v )] = 1 � D N ( � u � 2 2 )+ D N ( � v � 2 2 ) − D N ( � u − v � 2 � u , v ∈ ❘ N . 2 ) , 2 ◮ Overlap: N � u , v � N : = 1 u , v ∈ ❘ N . ∑ u i v i , N i = 1 ◮ Fix r ∈ [ 0; d ] : ❊ [ X N ( u ) X N ( v )] = D ( r ) − 1 2 D ( 2 ( r −� u , v � N )) , � u � 2 2 = � v � 2 2 = rN . ◮ ⇒ Localisation.

Particle in a rotationally invariant box ◮ Particle state space: √ S N : = { u ∈ ❘ N : � u � 2 ≤ L N } . ◮ A priori measure: µ N ∈ M finite ( S N ) : � � N d µ N ∑ u = ( u i ) N i = 1 ∈ ❘ N , ( u ) : = exp f ( u i ) , f : ❘ → ❘ , d λ N i = 1 f ( u ) : = h 1 u − h 2 u 2 , h 1 ∈ ❘ , h 2 ∈ ❘ + . ◮ Fyodorov, Sommers (2007): � � q max � r � C S ( β , r )[ x ] : = 1 d q q x ( s ) d s + h 2 log ( r − q max )+ 0 x ( q ) d q − h 2 r � r 1 2 0 � q max + β 2 � � D ′ ( 2 ( r − q max ))+ D ′ ( 2 ( r − q )) x ( q ) d q , 2 0 x ∈ X ( r ) .

A test function

Short range: solution of the variational problem Short range : D ( r ) : = B ( 0 ) − B ( r ) 3 [ D ′′′ ( r )] 2 / 2 − D ′′ ( r ) D ′′′ ( r ) = : S ( r ) > 0 , u ∈ ❘ + , Derrida’s random energy model (REM) behaviour: β ∈ [ 0; β c ) ⇒ RS optimiser β ∈ ( β c ; + ∞ ] ⇒ 1-RSB optimiser

Long range: solution of the variational problem Long range : if D satisfies S ( r ) < 0 , u ∈ ❘ + , Full RSB : β ∈ [ 0; β c ) ⇒ RS optimiser β ∈ ( β c ; + ∞ ] ⇒ FRSB optimiser

Critical range: logarithmic correlations Long range : D satisfies S ( r ) = 0 , u ∈ ❘ + , ◮ D ( r ) = log ( c + r ) , c > 0 ⇒ REM-behaviour (at the level of ❊ ) ◮ D ( r ) = ∑ n + 1 k = 0 K i log ( c i + r ) ( c 0 > c 1 > ... > c n + 1 and K i > 0 ) ⇒ generalised REM-behaviour ( n -RSB):

Sketch of proof Compare with a class of hierarchically correlated fields. ◮ “Generalised random energy model” : r ∈ [ 0; d ] : � � α ( 1 ) , α ( 2 ) ∈ ◆ n , a ( α ( 1 ) ) , a ( α ( 2 ) ) = − D ′ ( 2 ( r − q ( α ( 1 ) , α ( 2 ) ))) , Cov where 0 = q 0 < q 1 < ... < q n < q n + 1 = r , ultrametric overlap : q ( α ( 1 ) , α ( 2 ) ) : = q max { k ∈ [ 1; n ] ∩ ◆ : [ α ( 1 ) ] k =[ α ( 2 ) ] k } . ◮ Comparison process: N α ∈ ◆ n . ∑ A ( u , α ) : = u i a i ( α ) , u ∈ S N , i = 1

Multiplicative probability cascades ◮ Let { g i } 2 N i = 1 be i.i.d. r.v. of Gumbel extremal-type (say, Gaussian) 2 N ∑ δ exp ( u N ( g i ) / x ) = N → + ∞ ξ ( x ) , = = = ⇒ x ∈ ( 0;1 ) . i = 1 where u N ( y ) = a N y + b N is the linear extreme value normalisation . ◮ Ruelle (1987) : ξ ( x ) : = { δ ξ ( x ) i } i ∈ ◆ : = PPP ( ❘ + ∋ t �→ xt − x − 1 ) , x ∈ ( 0;1 )

Cascades Ruelle’s probability cascades (RPC) : ξ ( x 1 ) i 1 · ξ ( i 1 ) ( x 2 ) i 2 · ξ ( i 1 , i 2 ) ( x 3 ) i 3 ··· ξ ( i 1 , i 2 ,..., i n − 1 ) ( x n ) i n : i = ( i 1 ,..., i n ) ∈ ◆ n � � = : RPC ( x 1 ,..., x n ) , where 0 < x 1 < ... < x n < 1 , ξ ( i 1 , i 2 ,..., i k − 1 ) ( x k ) i.i.d. ξ ( x k ) .

RPC construction (sketch)

Comparison ◮ Interpolation : √ √ α ∈ ◆ n . H t ( u , α ) : = tX ( u )+ 1 − tA ( u , α ) , t ∈ [ 0;1 ] , u ∈ S N , ◮ Extended free energy functional : � � � ��� √ Φ N ( x )[ H t ] : = 1 � µ N ( d u ) ∑ β NH t ( u , α ) N ❊ log RPC ( x ) α exp . S N α ∈ ◆ n ◮ Fundamental theorem of calculus: � 1 d p N ( β ) = Φ N ( x )[ H 1 ] = Φ N ( x )[ H 0 ]+ d t Φ N ( x )[ H t ] d t , 0 where ◮ nonlinear summand = Φ N ( x )[ H 0 ] . � 1 d ◮ linear summand + (annoying) remainder = d t Φ N ( x )[ H t ] . 0