Top eigenvalue of a random matrix: Large deviations Satya N. - PowerPoint PPT Presentation

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��� ��� ��� ��� ��� ��� �� �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Large deviations and 3 -rd order phase transition typical fluctuations of size ∼ N − 2 / 3 → Tracy-Widom distributed Atypical rare fluctuations of size ∼ O (1) ⇒ not described by Tracy-Widom S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

��� ��� ��� ��� ��� ��� �� �� � � � � � � � � � � � � � � � � � � � � � � � � � � Large deviations and 3 -rd order phase transition typical fluctuations of size ∼ N − 2 / 3 → Tracy-Widom distributed Atypical rare fluctuations of size ∼ O (1) ⇒ not described by Tracy-Widom ⇒ rather by large deviation functions Pr( λ max ) typical TRACY−WIDOM −2/3 N large � � large (left) � � � � (right) � � � � � � 2 φ − e− N e− φ + � � N � � � � � � 2 λ max S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large deviations and 3 -rd order phase transition typical fluctuations of size ∼ N − 2 / 3 → Tracy-Widom distributed Atypical rare fluctuations of size ∼ O (1) ⇒ not described by Tracy-Widom ⇒ rather by large deviation functions Pr( λ max ) Pr( λ max ) critical point typical ��� ��� ��� ��� TRACY−WIDOM ��� ��� �� �� −2/3 � � N � � large large � � � � � � large large (left) (left) � � � � � � � � (right) � � (right) � � � � � � � � � � 2 φ − e− N 2 φ − � � e− φ + e− e− N φ + � � N � � N � � � � � � � � � � 2 2 λ max λ max S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large deviations and 3 -rd order phase transition typical fluctuations of size ∼ N − 2 / 3 → Tracy-Widom distributed Atypical rare fluctuations of size ∼ O (1) ⇒ not described by Tracy-Widom ⇒ rather by large deviation functions Pr( λ max ) Pr( λ max ) critical point typical ��� ��� ��� ��� TRACY−WIDOM ��� ��� �� �� −2/3 � � N � � large large � � � � � � large large (left) (left) � � � � � � � � (right) � � (right) � � � � � � � � � � 2 φ − e− N 2 φ − � � e− φ + e− e− N φ + � � N � � N � � � � � � � � � � 2 2 λ max λ max nonanalytic behavior of the large deviation functions √ at the critical point 2 = ⇒ 3 -rd order phase transition Review: S.M. & G. Schehr, J. Stat. Mech. P01012 (2014) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

II. Clue to phase transition S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stability of a Large Complex System S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Linear Stability of a Large Complex (Randomly Connected) System • Consider a stable non-interacting population of N species with equlibrium density ρ ⋆ i S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Linear Stability of a Large Complex (Randomly Connected) System • Consider a stable non-interacting population of N species with equlibrium density ρ ⋆ i Stable: x i = ρ i − ρ ⋆ i → small disturbed density dx i / dt = − x i → relaxes back to 0 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Linear Stability of a Large Complex (Randomly Connected) System • Consider a stable non-interacting population of N species with equlibrium density ρ ⋆ i Stable: x i = ρ i − ρ ⋆ i → small disturbed density dx i / dt = − x i → relaxes back to 0 • Now switch on the interaction between species S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Linear Stability of a Large Complex (Randomly Connected) System • Consider a stable non-interacting population of N species with equlibrium density ρ ⋆ i Stable: x i = ρ i − ρ ⋆ i → small disturbed density dx i / dt = − x i → relaxes back to 0 • Now switch on the interaction between species dx i / dt = − x i + α � N j =1 J ij x j J ij → ( N × N ) random interaction matrix α → interaction strength S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Linear Stability of a Large Complex (Randomly Connected) System • Consider a stable non-interacting population of N species with equlibrium density ρ ⋆ i Stable: x i = ρ i − ρ ⋆ i → small disturbed density dx i / dt = − x i → relaxes back to 0 • Now switch on the interaction between species dx i / dt = − x i + α � N j =1 J ij x j J ij → ( N × N ) random interaction matrix α → interaction strength • Question: What is the probabality that the system remains stable once the interaction is switched on? (R.M. May, Nature, 238, 413, 1972) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stability Criterion d • linear stability: dt [ x ] = [ α J − I ][ x ] ( J → random interaction matrix) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stability Criterion d • linear stability: dt [ x ] = [ α J − I ][ x ] ( J → random interaction matrix) Let { λ 1 , λ 2 , · · · , λ N } → eigenvalues of the matrix J S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stability Criterion d • linear stability: dt [ x ] = [ α J − I ][ x ] ( J → random interaction matrix) Let { λ 1 , λ 2 , · · · , λ N } → eigenvalues of the matrix J • Stable if αλ i < 1 for all i = 1 , 2 , · · · , N ⇒ λ max < 1 α = w → stability criterion w → inverse interaction strength S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stability Criterion d • linear stability: dt [ x ] = [ α J − I ][ x ] ( J → random interaction matrix) Let { λ 1 , λ 2 , · · · , λ N } → eigenvalues of the matrix J • Stable if αλ i < 1 for all i = 1 , 2 , · · · , N ⇒ λ max < 1 α = w → stability criterion w → inverse interaction strength • Prob.(the system is stable)=Prob.[ λ max < w ] = P ( w , N ) Cumulative distribution of the top eigenvalue S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stable-Unstable Phase Transition as N → ∞ • Assuming that the interaction matrix J ij → Real Symmetric Gaussian � � − N i , j J 2 � − N 2 Tr ( J 2 ) � Prob . [ J ij ] ∝ exp � ∝ exp ij 2 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Stable-Unstable Phase Transition as N → ∞ • Assuming that the interaction matrix J ij → Real Symmetric Gaussian � � − N i , j J 2 � − N 2 Tr ( J 2 ) � Prob . [ J ij ] ∝ exp � ∝ exp ij 2 • May observed a sharp phase transition as N → ∞ : √ w = 1 α > 2 ⇒ Stable (weakly interacting) √ w = 1 α < 2 ⇒ Unstable (strongly interacting) Prob.(the system is stable)=Prob.[ λ max < w ] = P ( w , N ) w P( ) , N STABLE 1 0 w UNSTABLE 2 = 1/ α S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Finite but Large N : Prob.(the system is stable)=Prob.[ λ max < w ] = P ( w , N ) What happens for finite but large N ? S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Finite but Large N : Prob.(the system is stable)=Prob.[ λ max < w ] = P ( w , N ) What happens for finite but large N ? w = Prob.[ λ max < w ] P( , ) N STABLE 1 finite but large N 0 2 UNSTABLE w • Is there any thermodynamic sense to this phase transition? • What is the analogue of free energy? • What is the order of this phase transition? S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

III. Summary of Results S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

For Large but Finite N : Summary of Results P( w , ) = Prob.[ λ max < w ] N STABLE 1 width of O ( N −2/3 ) finite but large N 2 w UNSTABLE S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

For Large but Finite N : Summary of Results P( w , ) = Prob.[ λ max < w ] N STABLE 1 width of O ( N −2/3 ) finite but large N 2 w UNSTABLE √ − N 2 Φ − ( w ) + . . . � � P ( w , N ) ∼ exp for 2 − w ∼ O (1) � √ √ √ 2 N 2 / 3 � �� 2 | ∼ O ( N − 2 / 3 ) ∼ F 1 w − 2 for | w − √ ∼ 1 − exp [ − N Φ + ( w ) + . . . ] for w − 2 ∼ O (1) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

For Large but Finite N : Summary of Results P( w , ) = Prob.[ λ max < w ] N STABLE 1 width of O ( N −2/3 ) finite but large N 2 w UNSTABLE √ − N 2 Φ − ( w ) + . . . � � P ( w , N ) ∼ exp for 2 − w ∼ O (1) � √ √ √ 2 N 2 / 3 � �� 2 | ∼ O ( N − 2 / 3 ) ∼ F 1 w − 2 for | w − √ ∼ 1 − exp [ − N Φ + ( w ) + . . . ] for w − 2 ∼ O (1) Crossover function: F 1 ( z ) → Tracy-Widom (1994) Exact rate functions: Φ − ( w ) → Dean & S.M. 2006 Φ + ( w ) → S.M. & Vergassola 2009 Higher order corrections: ( Borot, Eynard, S.M., & Nadal 2011, Nadal & S.M., 2011 ) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N : S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N : • Left large deviation function: 1 � 36 w 2 − w 4 − (15 w + w 3 ) � w 2 + 6 Φ − ( w ) = 108 √ � �� � + 27 ln(18) − 2 ln( w + 6 + w 2 ) w < 2 where [D. S. Dean & S.M., PRL, 97, 160201 (2006)] S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N : • Left large deviation function: 1 � 36 w 2 − w 4 − (15 w + w 3 ) � w 2 + 6 Φ − ( w ) = 108 √ � �� � + 27 ln(18) − 2 ln( w + 6 + w 2 ) w < 2 where [D. S. Dean & S.M., PRL, 97, 160201 (2006)] √ √ 1 2 − w ) 3 In particular, as w → 2 (from left), Φ − ( w ) → 2 ( √ 6 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N : • Left large deviation function: 1 � 36 w 2 − w 4 − (15 w + w 3 ) � w 2 + 6 Φ − ( w ) = 108 √ � �� � + 27 ln(18) − 2 ln( w + 6 + w 2 ) w < 2 where [D. S. Dean & S.M., PRL, 97, 160201 (2006)] √ √ 1 2 − w ) 3 In particular, as w → 2 (from left), Φ − ( w ) → 2 ( √ 6 • Right large deviation function: √ � � w 2 − 2 √ Φ + ( w ) = 1 w − � w 2 − 2 + ln √ w > 2 2 w where 2 [S.M. & Vergassola, PRL, 102, 060601 (2009)] S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Left and Right Large Deviation Function Using Coulomb gas + Saddle point method for large N : • Left large deviation function: 1 � 36 w 2 − w 4 − (15 w + w 3 ) � w 2 + 6 Φ − ( w ) = 108 √ � �� � + 27 ln(18) − 2 ln( w + 6 + w 2 ) w < 2 where [D. S. Dean & S.M., PRL, 97, 160201 (2006)] √ √ 1 2 − w ) 3 In particular, as w → 2 (from left), Φ − ( w ) → 2 ( √ 6 • Right large deviation function: √ � � w 2 − 2 √ Φ + ( w ) = 1 w − � w 2 − 2 + ln √ w > 2 2 w where 2 [S.M. & Vergassola, PRL, 102, 060601 (2009)] √ √ Φ + ( w ) → 2 7 / 4 2) 3 / 2 As w → 2 (from right), 3 ( w − S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large Deviation Functions These large deviation functions Φ ± ( w ) have been found useful in a large variety of problems: [ Fyodorov 2004, Fyodorov & Williams 2007, Bray & Dean 2007, Auffinger, Ben Arous & Cerny 2010, Fydorov & Nadal 2012.... —— stationary points on random Gaussian surfaces and spin glass landscapes] [ Cavagna, Garrahan, Giardina 2000,... —— Glassy systems] [ Susskind 2003, Douglas et. al. 2004, Aazami & Easther 2006, Marsh et. al. 2011, ... —— String theory & Cosmology] [ Beltrani 2007, Dedieu & Malajovich, 2007, Houdre 2011... ——Random Polynomials, Random Words (Young diagrams) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

3-rd Order Phase Transition √  � − N 2 Φ − ( w ) + . . . � exp for w < 2 ( unstable )  P ( w , N ) ≈ √ 1 − exp {− N Φ + ( w ) + . . . } for w > 2 ( stable )  S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

3-rd Order Phase Transition √  � − N 2 Φ − ( w ) + . . . � exp for w < 2 ( unstable )  P ( w , N ) ≈ √ 1 − exp {− N Φ + ( w ) + . . . } for w > 2 ( stable )  √ √  − 2 − w ) 3 Φ − ( w ) ∼ ( as w → 2 N →∞ − 1   lim N 2 ln [ P ( w , N )] = √ +  0 as w → 2  − → analogue of the free energy difference S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

3-rd Order Phase Transition √  � − N 2 Φ − ( w ) + . . . � exp for w < 2 ( unstable )  P ( w , N ) ≈ √ 1 − exp {− N Φ + ( w ) + . . . } for w > 2 ( stable )  √ √  − 2 − w ) 3 Φ − ( w ) ∼ ( as w → 2 N →∞ − 1   lim N 2 ln [ P ( w , N )] = √ +  0 as w → 2  − → analogue of the free energy difference Tracy−Widom [−ln P]/N 2 large) finite N ( w 2 N limit ~ ( 2 _ w) 3 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

3-rd Order Phase Transition √  � − N 2 Φ − ( w ) + . . . � exp for w < 2 ( unstable )  P ( w , N ) ≈ √ 1 − exp {− N Φ + ( w ) + . . . } for w > 2 ( stable )  √ √  − 2 − w ) 3 Φ − ( w ) ∼ ( as w → 2 N →∞ − 1   lim N 2 ln [ P ( w , N )] = √ +  0 as w → 2  − → analogue of the free energy difference 3-rd derivative → discontinuous Tracy−Widom [−ln P]/N 2 √ Crossover: N → ∞ , w → 2 keeping large) finite N ( √ 2) N 2 / 3 fixed ( w − � √ √ 2 N 2 / 3 � �� w P ( w , N ) → F 1 w − 2 2 N limit ~ ( 2 _ w) 3 → Tracy-Widom S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large N Phase Transition: Phase Diagram 1 N crossover STABLE UNSTABLE ( weakly interacting ) strongly interacting ) ( �� �� �� �� 0 α= 1 w 1 2 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

A nice review of large-N gauge theory: M. Marino, arXiv:1206.6272 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

�� �� �� �� Large N Phase Transition: Phase Diagram U(N) lattice gauge theory in 2−d GROSS−WITTEN−WADIA transition (1980) 1 N crossover WEAK STRONG �� �� �� �� 0 g c coupling strength g S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large N Phase Transition: Phase Diagram U(N) lattice gauge theory in 2−d GROSS−WITTEN−WADIA transition (1980) 1 N 1 N crossover crossover STABLE UNSTABLE WEAK STRONG ( weakly interacting ) ( strongly interacting ) �� �� �� �� �� �� �� �� 0 g c 0 α = 1 w 1 2 coupling strength g S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large N Phase Transition: Phase Diagram U(N) lattice gauge theory in 2−d GROSS−WITTEN−WADIA transition (1980) 1 N 1 N crossover crossover STABLE UNSTABLE WEAK STRONG ( weakly interacting ) ( strongly interacting ) �� �� �� �� �� �� �� �� 0 g c 0 α = 1 w 1 2 coupling strength g Similar 3-rd order phase transition in U ( N ) lattice-gauge theory in 2-d Unstable phase ≡ Strong coupling phase of Yang-Mills gauge theory Stable phase ≡ Weak coupling phase of Yang-Mills gauge theory Tracy-Widom ⇒ crossover function in the double scaling regime (for finite but large N ) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Main Conclusion: Tracy-Widom distribution is a universal crossover function associated with a 3-rd order phase transition Review: S.M. & G. Schehr, J. Stat. Mech. P01012 (2014) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

IV. Coulomb Gas S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Gaussian Random Matrices • ( N × N ) Gaussian random matrix: J ≡ [ J ij ] • Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Gaussian Random Matrices • ( N × N ) Gaussian random matrix: J ≡ [ J ij ] • Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) � � − β � J † J � • Prob [ J ij ] ∝ exp 2 N Tr ) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Gaussian Random Matrices • ( N × N ) Gaussian random matrix: J ≡ [ J ij ] • Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) � � − β � J † J � • Prob [ J ij ] ∝ exp 2 N Tr ) • N real eigenvalues { λ 1 , λ 2 , . . . , λ N } → correlated random variables S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Gaussian Random Matrices • ( N × N ) Gaussian random matrix: J ≡ [ J ij ] • Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) � � − β � J † J � • Prob [ J ij ] ∝ exp 2 N Tr ) • N real eigenvalues { λ 1 , λ 2 , . . . , λ N } → correlated random variables • Joint distribution of eigenvalues ( Wigner, 1951 ) � N � � P ( λ 1 , λ 2 , . . . , λ N ) = 1 − β � λ 2 | λ j − λ k | β exp 2 N i Z N i =1 j < k where the Dyson index β = 1 (GOE), β = 2 (GUE) or β = 4 (GSE) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Gaussian Random Matrices • ( N × N ) Gaussian random matrix: J ≡ [ J ij ] • Ensembles: Orthogonal (GOE), Unitary (GUE) or Symplectic (GSE) � � − β � J † J � • Prob [ J ij ] ∝ exp 2 N Tr ) • N real eigenvalues { λ 1 , λ 2 , . . . , λ N } → correlated random variables • Joint distribution of eigenvalues ( Wigner, 1951 ) � N � � P ( λ 1 , λ 2 , . . . , λ N ) = 1 − β � λ 2 | λ j − λ k | β exp 2 N i Z N i =1 j < k where the Dyson index β = 1 (GOE), β = 2 (GUE) or β = 4 (GSE) • Z N = Partition Function � ∞ � ∞ � N � � − β � � λ 2 | λ j − λ k | β = . . . { d λ i } exp 2 N i −∞ −∞ i i =1 j < k S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �������������������� �������������������� � � Coulomb Gas Interpretation • Z N =     � ∞ � ∞ N  − β   � � N λ 2 � . . . { d λ i } exp i − log | λ j − λ k |  2 −∞ −∞   i i =1 j � = k S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Coulomb Gas Interpretation • Z N =     � ∞ � ∞ N  − β   � � N λ 2 � . . . { d λ i } exp i − log | λ j − λ k |  2 −∞ −∞   i i =1 j � = k • 2-d Coulomb gas confined to a line (Dyson) with β → inverse temp. λ 1 λ 2 λ 3 λ Ν � � � � � � � � confining � � � � � � parabolic � � � � potential � � � � � � � � � � � � � � � � �������������������� �������������������� � � 0 λ S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Coulomb Gas Interpretation • Z N =     � ∞ � ∞ N  − β   � � N λ 2 � . . . { d λ i } exp i − log | λ j − λ k |  2 −∞ −∞   i i =1 j � = k • 2-d Coulomb gas confined to a line (Dyson) with β → inverse temp. λ 1 λ 2 λ 3 λ Ν � � � � � � � � confining � � � � � � parabolic � � � � potential � � � � � � � � � � � � � � � � �������������������� �������������������� � � 0 λ • Balance of energy ⇒ N 2 λ 2 ∼ N 2 • Typical eigenvalue: λ typ ∼ O (1) for large N S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Spectral Density: Wigner’s Semicircle Law N • Av. density of states: ρ ( λ, N ) = � 1 � δ ( λ − λ i ) � N i =1 N →∞ ρ ( λ ) = 1 � 2 − λ 2 • Wigner’s Semi-circle: ρ ( λ, N ) − − − − → π ρ(λ) WIGNER SEMI−CIRCLE SEA − 2 0 2 λ √ • � λ max � = 2 for large N . • λ max fluctuates from one sample to another. Prob[ λ max , N ] = ? S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Probability of Large Deviations of λ max : TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 N LEFT LARGE DEVIATION − 2 0 2 λ RIGHT LARGE DEVIATION � √ √ 2 N 2 / 3 ( w − � • Tracy-Widom law Prob [ λ max ≤ w , N ] → F β 2) describes the prob. of typical (small) fluctuations of ∼ O ( N − 2 / 3 ) √ √ 2 | ∼ N − 2 / 3 around the mean 2, i.e., when | λ max − S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Probability of Large Deviations of λ max : TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 N LEFT LARGE DEVIATION − 2 0 2 λ RIGHT LARGE DEVIATION � √ √ 2 N 2 / 3 ( w − � • Tracy-Widom law Prob [ λ max ≤ w , N ] → F β 2) describes the prob. of typical (small) fluctuations of ∼ O ( N − 2 / 3 ) √ √ 2 | ∼ N − 2 / 3 around the mean 2, i.e., when | λ max − • Q: How to describe the prob. of large (atypical) fluctuations when √ | λ max − 2 | ∼ O (1) → Large deviations from mean S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Large Deviation Tails of λ max TRACY−WIDOM ρ (λ, Ν) WIGNER SEMI−CIRCLE −2/3 N LEFT LARGE DEVIATION − 2 0 2 λ RIGHT LARGE DEVIATION Prob. density of the top eigenvalue: Prob . [ λ max = w , N ] behaves as: √ − β N 2 Φ − ( w ) � � ∼ exp for 2 − w ∼ O (1) � √ √ √ 2 N 2 / 3 � �� N 2 / 3 f β 2 | ∼ O ( N − 2 / 3 ) ∼ w − 2 for | w − √ ∼ exp [ − β N Φ + ( w )] for w − 2 ∼ O (1) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

V. Saddle Point Method S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Distribution of λ max : Saddle Point Method Prob [ λ max ≤ w , N ] = Prob [ λ 1 ≤ w , λ 2 ≤ w , . . . , λ N ≤ w ] = Z N ( w ) Z N ( ∞ )     � w � w N  − β   � � λ 2 � Z N ( w ) = . . . { d λ i } exp  N i − log | λ j − λ k |  2 −∞ −∞ i =1 j � = k  i S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Distribution of λ max : Saddle Point Method Prob [ λ max ≤ w , N ] = Prob [ λ 1 ≤ w , λ 2 ≤ w , . . . , λ N ≤ w ] = Z N ( w ) Z N ( ∞ )     � w � w N  − β   � � λ 2 � Z N ( w ) = . . . { d λ i } exp  N i − log | λ j − λ k |  2 −∞ −∞ i =1 j � = k  i denominator numerator WALL w λ λ S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Distribution of λ max : Saddle Point Method Prob [ λ max ≤ w , N ] = Prob [ λ 1 ≤ w , λ 2 ≤ w , . . . , λ N ≤ w ] = Z N ( w ) Z N ( ∞ )     � w � w N  − β   � � λ 2 � Z N ( w ) = . . . { d λ i } exp  N i − log | λ j − λ k |  2 −∞ −∞ i =1 j � = k  i denominator numerator WALL w λ λ S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Setting up the Saddle Point Method � w � − β N 2 E ( { λ i } ) � � • Z N ( w ) ∝ d λ i exp −∞ i E ( { λ i } ) = 1 1 � λ 2 � i − log | λ j − λ k | 2 N 2 2 N j � = k i S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Setting up the Saddle Point Method � w � − β N 2 E ( { λ i } ) � � • Z N ( w ) ∝ d λ i exp −∞ i E ( { λ i } ) = 1 1 � λ 2 � i − log | λ j − λ k | 2 N 2 2 N j � = k i • As N → ∞ → discrete sum → continuous integral: �� w � w � w E [ ρ ( λ )] = 1 � λ 2 ρ ( λ ) d λ − ln | λ − λ ′ | ρ ( λ ) ρ ( λ ′ ) d λ d λ ′ 2 −∞ −∞ −∞ S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Setting up the Saddle Point Method � w � − β N 2 E ( { λ i } ) � � • Z N ( w ) ∝ d λ i exp −∞ i E ( { λ i } ) = 1 1 � λ 2 � i − log | λ j − λ k | 2 N 2 2 N j � = k i • As N → ∞ → discrete sum → continuous integral: �� w � w � w E [ ρ ( λ )] = 1 � λ 2 ρ ( λ ) d λ − ln | λ − λ ′ | ρ ( λ ) ρ ( λ ′ ) d λ d λ ′ 2 −∞ −∞ −∞ where the charge density: ρ ( λ ) = 1 � i δ ( λ − λ i ) N S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Setting up the Saddle Point Method � w � − β N 2 E ( { λ i } ) � � • Z N ( w ) ∝ d λ i exp −∞ i E ( { λ i } ) = 1 1 � λ 2 � i − log | λ j − λ k | 2 N 2 2 N j � = k i • As N → ∞ → discrete sum → continuous integral: �� w � w � w E [ ρ ( λ )] = 1 � λ 2 ρ ( λ ) d λ − ln | λ − λ ′ | ρ ( λ ) ρ ( λ ′ ) d λ d λ ′ 2 −∞ −∞ −∞ where the charge density: ρ ( λ ) = 1 � i δ ( λ − λ i ) N � � � �� �� � − β N 2 Z N ( w ) ∝ D ρ ( λ ) exp E [ ρ ( λ )] + C ρ ( λ ) d λ − 1 + O ( N ) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Setting up the Saddle Point Method � w � − β N 2 E ( { λ i } ) � � • Z N ( w ) ∝ d λ i exp −∞ i E ( { λ i } ) = 1 1 � λ 2 � i − log | λ j − λ k | 2 N 2 2 N j � = k i • As N → ∞ → discrete sum → continuous integral: �� w � w � w E [ ρ ( λ )] = 1 � λ 2 ρ ( λ ) d λ − ln | λ − λ ′ | ρ ( λ ) ρ ( λ ′ ) d λ d λ ′ 2 −∞ −∞ −∞ where the charge density: ρ ( λ ) = 1 � i δ ( λ − λ i ) N � � � �� �� � − β N 2 Z N ( w ) ∝ D ρ ( λ ) exp E [ ρ ( λ )] + C ρ ( λ ) d λ − 1 + O ( N ) � • for large N , minimize the action S [ ρ ( λ )] = E [ ρ ( λ )] + C [ ρ ( λ ) d λ − 1] δ S Saddle Point Method: δρ = 0 ⇒ ρ w ( λ ) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Setting up the Saddle Point Method � w � − β N 2 E ( { λ i } ) � � • Z N ( w ) ∝ d λ i exp −∞ i E ( { λ i } ) = 1 1 � λ 2 � i − log | λ j − λ k | 2 N 2 2 N j � = k i • As N → ∞ → discrete sum → continuous integral: �� w � w � w E [ ρ ( λ )] = 1 � λ 2 ρ ( λ ) d λ − ln | λ − λ ′ | ρ ( λ ) ρ ( λ ′ ) d λ d λ ′ 2 −∞ −∞ −∞ where the charge density: ρ ( λ ) = 1 � i δ ( λ − λ i ) N � � � �� �� � − β N 2 Z N ( w ) ∝ D ρ ( λ ) exp E [ ρ ( λ )] + C ρ ( λ ) d λ − 1 + O ( N ) � • for large N , minimize the action S [ ρ ( λ )] = E [ ρ ( λ )] + C [ ρ ( λ ) d λ − 1] δ S Saddle Point Method: δρ = 0 ⇒ ρ w ( λ ) � − β N 2 S [ ρ w ( λ )] � ⇒ Z N ( w ) ∼ exp S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Saddle Point Solution • saddle point δ S δρ = 0 ⇒ � w λ 2 − 2 ρ w ( λ ′ ) ln | λ − λ ′ | d λ ′ + C = 0 −∞ S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Saddle Point Solution • saddle point δ S δρ = 0 ⇒ � w λ 2 − 2 ρ w ( λ ′ ) ln | λ − λ ′ | d λ ′ + C = 0 −∞ • Taking a derivative w.r.t. λ gives a singular integral Eq. � w ρ w ( λ ′ ) d λ ′ λ = P for λ ∈ [ −∞ , w ] → Semi-Hilbert transform λ − λ ′ −∞ − → force balance condition S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Saddle Point Solution • saddle point δ S δρ = 0 ⇒ � w λ 2 − 2 ρ w ( λ ′ ) ln | λ − λ ′ | d λ ′ + C = 0 −∞ • Taking a derivative w.r.t. λ gives a singular integral Eq. � w ρ w ( λ ′ ) d λ ′ λ = P for λ ∈ [ −∞ , w ] → Semi-Hilbert transform λ − λ ′ −∞ − → force balance condition • When w → ∞ : solution − → Wigner semi-circle law √ ρ ∞ ( λ ) = 1 2 − λ 2 π S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Saddle Point Solution • saddle point δ S δρ = 0 ⇒ � w λ 2 − 2 ρ w ( λ ′ ) ln | λ − λ ′ | d λ ′ + C = 0 −∞ • Taking a derivative w.r.t. λ gives a singular integral Eq. � w ρ w ( λ ′ ) d λ ′ λ = P for λ ∈ [ −∞ , w ] → Semi-Hilbert transform λ − λ ′ −∞ − → force balance condition • When w → ∞ : solution − → Wigner semi-circle law √ ρ ∞ ( λ ) = 1 2 − λ 2 π Exact solution for all w : [D. S. Dean & S.M., PRL, 97, 160201 (2006); PRE, 77, 041108 (2008)] S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

� � � � �� �� �� �� Exact Saddle Point Solution • Exact solution ( D. Dean and S.M., 2006, 2008 ): √ √  1 2 − λ 2 for w ≥ 2 π   ρ w ( λ ) = √ √ λ + L ( w )  2 π √ w − λ [ w + L ( w ) − 2 λ ] for w < 2  √ w 2 + 6 − w ] / 3 where L ( w ) = [2 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Saddle Point Solution • Exact solution ( D. Dean and S.M., 2006, 2008 ): √ √  1 2 − λ 2 for w ≥ 2 π   ρ w ( λ ) = √ √ λ + L ( w )  2 π √ w − λ [ w + L ( w ) − 2 λ ] for w < 2  √ w 2 + 6 − w ] / 3 where L ( w ) = [2 ρ w (λ) vs. λ for different charge density W W > W < 2 2 W= 2 w w w � � � � �� �� �� �� − w − − L(w) 2 2 2 2 2 pushed critical unpushed (STABLE) (UNSTABLE) W= 2 CRITICAL POINT S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Exact Saddle Point Solution • Exact solution ( D. Dean and S.M., 2006, 2008 ): √ √  1 2 − λ 2 for w ≥ 2 π   ρ w ( λ ) = √ √ λ + L ( w )  2 π √ w − λ [ w + L ( w ) − 2 λ ] for w < 2  √ w 2 + 6 − w ] / 3 where L ( w ) = [2 ρ w (λ) vs. λ for different charge density W W > W < 2 2 W= 2 w w w � � � � �� �� �� �� − w − − L(w) 2 2 2 2 2 pushed critical unpushed (STABLE) (UNSTABLE) W= 2 CRITICAL POINT S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Left Large Deviation Function Prob [ λ max ≤ w , N ] = Z N ( w ) − β N 2 { S [ ρ w ( λ )] − S [ ρ ∞ ( λ )] } � � ∼ exp Z N ( ∞ ) − β N 2 Φ − ( w ) � � ∼ exp S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Left Large Deviation Function Prob [ λ max ≤ w , N ] = Z N ( w ) − β N 2 { S [ ρ w ( λ )] − S [ ρ ∞ ( λ )] } � � ∼ exp Z N ( ∞ ) − β N 2 Φ − ( w ) � � ∼ exp N →∞ − 1 lim N 2 ln [ P ( w , N )] = Φ − ( w ) → left large deviation function physically Φ − ( w ) − → energy cost in pushing the Coulomb gas S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Left Large Deviation Function Prob [ λ max ≤ w , N ] = Z N ( w ) − β N 2 { S [ ρ w ( λ )] − S [ ρ ∞ ( λ )] } � � ∼ exp Z N ( ∞ ) − β N 2 Φ − ( w ) � � ∼ exp N →∞ − 1 lim N 2 ln [ P ( w , N )] = Φ − ( w ) → left large deviation function physically Φ − ( w ) − → energy cost in pushing the Coulomb gas 1 � 36 w 2 − w 4 − (15 w + w 3 ) � w 2 + 6 Φ − ( w ) = 108 √ � �� � 6 + w 2 ) + 27 ln(18) − 2 ln( w + for w < 2 ( Dean & S.M., 2006,2008 ) S.N. Majumdar Top eigenvalue of a random matrix: Large deviations

Left Large Deviation Function Prob [ λ max ≤ w , N ] = Z N ( w ) − β N 2 { S [ ρ w ( λ )] − S [ ρ ∞ ( λ )] } � � ∼ exp Z N ( ∞ ) − β N 2 Φ − ( w ) � � ∼ exp N →∞ − 1 lim N 2 ln [ P ( w , N )] = Φ − ( w ) → left large deviation function physically Φ − ( w ) − → energy cost in pushing the Coulomb gas 1 � 36 w 2 − w 4 − (15 w + w 3 ) � w 2 + 6 Φ − ( w ) = 108 √ � �� � 6 + w 2 ) + 27 ln(18) − 2 ln( w + for w < 2 ( Dean & S.M., 2006,2008 ) √ √ 2 − w ) 3 as w → 1 Note also that Φ − ( w ) ≈ 2 ( 2 from below √ 6 S.N. Majumdar Top eigenvalue of a random matrix: Large deviations