Results Consider the case when k , β ∈ N . Then k β 2 ≥ 1 so we expect MoM N ( k , β ) ∼ ρ k ,β N k 2 β 2 − k +1 . Theorem [B.-Keating (2018)] Let k , β ∈ N . Then MoM N ( k , β ) is a polynomial in N . Theorem [B.-Keating (2018)] Let k , β ∈ N . Then with ρ k ,β an explicit function of k and β , MoM N ( k , β ) = ρ k ,β N k 2 β 2 − k +1 + O ( N k 2 β 2 − k ) . Emma Bailey Moments of Moments CIRM 2019 4 / 6
Example MoM N (2 , 3) = ( N +1)( N +2)( N +3)( N +4)( N +5)( N +6)( N +7)( N +8)( N +9)( N +10)( N +11) 1722191327731024154944441889587200000000 � 12308743625763 N 24+1772459082109872 N 23+121902830804059138 N 22+ × +5328802119564663432 N 21+166214570195622478453 N 20+3937056259812505643352 N 19 +73583663800226157619008 N 18+1113109355823972261429312 N 17+13869840005250869763713293 N 16 +144126954435929329947378912 N 15+1259786144898207172443272698 N 14 +9315726913410827893883025672 N 13+58475127984013141340467825323 N 12 +311978271286536355427593012632 N 11+1413794106539529439589778645028 N 10 +5427439874579682729570383266992 N 9+17564370687865211818995713096848 N 8 +47561382824003032731805262975232 N 7+106610927256886475209611301000128 N 6 +194861499503272627170466392014592 N 5+284303877221735683573377603640320 N 4 +320989495108428049992898521600000 N 3+266974288159876385845370793984000 N 2 � +148918006780282798012340305920000 N +43144523802785397500411904000000 Emma Bailey Moments of Moments CIRM 2019 5 / 6
Thank you Emma Bailey Moments of Moments CIRM 2019 6 / 6
Speed of Convergence in the Gaussian Distribution for Laguerre Ensembles Under Double Scaling Sergey Berezin 1 and Alexander Bufetov 2 1 CNRS, PDMI RAS, 2 CNRS, Steklov IITP RAS E-mail: 1 servberezin@yandex.ru, 2 bufetov@mi-ras.ru
Problem statement Consider Laguerre Unitary Ensemble : M = U ∗ diag { Λ 1 , . . . , Λ n } U, (1) where U is distributed uniformly on the unitary group U ( n ) The random variables Λ 1 , . . . , Λ n have the joint probability density 1 ∏︂ j e − 4 mλ j ∏︂ 1 [ λ j > 0] λ α ( λ k − λ j ) 2 , P n,m ( λ 1 , . . . , λ n ) = (2) Z n,m j j<k where α > − 1 , m ∈ N , and Z n,m is the partition function. Let f ( M ) be a real-valued function defined on the spectrum of M . Our goal is to study the characteristic function ∫︂ e ih ∑︁ f ( λ j ) P n,m ( λ 1 , . . . , λ n ) dλ 1 · . . . · dλ n e ih Tr f ( M ) ]︁ [︁ = (3) E n,m of the linear statistic Tr f ( M ) in a double-scaling limit as n = m → ∞ . Conference “Integrability and Randomness in Mathematical Physics and Geometry”, April 8–12, 2019 1
Main results Let f : R + → R be locally H¨ older continuous such that it admits the analytic continuation to some neighborhood of [0 , 1] . Theorem (Convergence to the Gaussian law) d Tr f ( M ) − n κ [ f ] − → N ( µ [ f ] , K [ f ]) , n = m → ∞ . (4) The linear functionals κ [ f ] , µ [ f ] , and the quadratic functional K [ f ] are given with the explicit formulas. Theorem (Speed of convergence) Let f ( x ) also satisfy f ( x ) = O ( e Ax ) , A > 0 , as x → + ∞ . Define the cumulative distribution functions F n ( x ) and F ( x ) corresponding to Tr f ( M ) − n κ [ f ] − µ [ f ] and to N (0 , K [ f ]) , respectively. Then sup x | F n ( x ) − F ( x ) | = O (1 /n ) , n = m → ∞ . (5) Conference “Integrability and Randomness in Mathematical Physics and Geometry”, April 8–12, 2019 2
The proof of Theorems is based on the Riemann–Hilbert analysis similar to Charlier&Gharakhloo (2019). However, unlike them, we are interested in complex exponents . In such a case the corresponding Hankel determinants and/or the weight of the corresponding orthogonal polynomials can be zero. Also we need the exponents that grow with n . To succeed we adopt the approach from Deift, Its&Krasovsky (2014) and use the deformation of w ( x ) = x α e − 4 nx e ihf ( x ) . (6) into w l,t ( x ) = x α e − 4 nx (︁ 1 − t + te ih 1 [ l<n γ +1] f ( x ) )︁ e ih ( l − 1) 1 [ l<n γ +1] f ( x ) , ˜ (7) We choose ε > 0 small enough so that 1 − t + te ih 1 [ l<n γ +1] f ( x ) ̸ = 0 , γ ∈ [0 , 1] , (8) for all t ∈ [0 , 1] , x in the neighborhood of [0 , 1] , h such that | h | < ε , and for all n, l . Conference “Integrability and Randomness in Mathematical Physics and Geometry”, April 8–12, 2019 3
From Gumbel to Tracy–Widom II, via integer partitions Dan Betea University of Bonn based on joint work with J. Bouttier (Math. Phys. Anal. Geom. 2019, arXiv:1807.09022 [math-ph] ) CIRM 1X.IV.MM19
Partitions • • ◦ • • ◦ • • • ◦ ◦ ◦ ◦ ◦ Figure: Partition (Young diagram) λ = (2 , 2 , 2 , 1 , 1) (Frobenius coordinates (1 , 0 | 4 , 1)) in English, French and Russian notation, with associated Maya diagram (particle-hole representation). Size | λ | = 8, length ℓ ( λ ) = 5. Figure: Skew partitions (Young diagrams) (4 , 3 , 2 , 1) / (2 , 1) (but also (5 , 4 , 3 , 2 , 1) / (5 , 2 , 1) , . . . ) and (4 , 4 , 2 , 1) / (2 , 2) (but also (6 , 4 , 4 , 2 , 1) / (6 , 2 , 2) , . . . )
Counting tableaux A standard Young tableau (SYT) is a filling of a (possibly skew) Young diagram with numbers 1 , 2 , . . . strictly increasing down columns and rows. 1 3 5 6 1 7 2 4 9 3 4 7 2 5 8 6 dim λ := number of SYTs of shape λ and similarly for dim λ/µ .
Measures on partitions There are two natural measures on all partitions: poissonized Plancherel vs. (grand canonical) uniform Prob ( λ ) = e − ǫ 2 ǫ 2 | λ | (dim λ ) 2 Prob ( λ ) = u | λ | � (1 − u i ) vs. ( | λ | !) 2 i ≥ 1 with ǫ ≥ 0 and 1 > u ≥ 0 parameters.
Ulam’s problem and Hammersley last passage percolation Quantity of interest: L = longest up-right path from (0 , 0) to (1 , 1) (= 4 here). Schensted’s theorem yields that, in distribution, L = λ 1 with λ coming from the poissonized Plancherel measure.
The Baik–Deift–Johansson theorem and Tracy–Widom Theorem (BaiDeiJoh 1999) If λ is distributed as poissonized Plancherel, we have: � λ 1 − 2 ǫ � ǫ →∞ Prob lim ≤ s = F TW ( x ) := det(1 − Ai 2 ) ( s , ∞ ) ǫ 1 / 3 with � ∞ Ai 2 ( x , y ) := Ai ( x + s ) Ai ( y + s ) ds . 0 and Ai the Airy function (solution of y ′′ = xy decaying at ∞ ). F TW is the Tracy–Widom GUE distribution. It is by (original) construction the extreme distribution of the largest eigenvalue of a random hermitian matrix with iid standard Gaussian entries as the size of the matrix goes to infinity.
The Erd˝ os–Lehner theorem and Gumbel Theorem (ErdLeh 1941) For the uniform measure Prob ( λ ) ∝ u | λ | we have: � λ 1 < − log(1 − u ) ξ � = e − e − ξ . u → 1 − Prob lim + log u | log u |
The finite temperature Plancherel measure On pairs of partitions µ ⊂ λ ⊃ µ consider the measure Prob ( µ, λ ) ∝ u | µ | · ǫ 2( | λ |−| µ | ) dim 2 ( λ/µ ) ( | λ/µ | !) 2 with u = e − β , β = inverse temperature. ◮ u = 0 yields the poissonized Plancherel measure ◮ ǫ = 0 yields the (grand canonical) uniform measure
The finite temperature Plancherel measure II Theorem (B/Bouttier 2019) ǫ 1 − u → ∞ and u = exp( − α M − 1 / 3 ) → 1 . Then Let M = � λ 1 − 2 M � = F α ( x ) := det(1 − Ai α ) ( s , ∞ ) M →∞ Prob lim ≤ s M 1 / 3 with � ∞ e α s Ai α ( x , y ) := 1 + e α s · Ai ( x + s ) Ai ( y + s ) ds . −∞ the finite temperature Airy kernel.
What is in a part? PPP ( u 4 ǫ 2 ) PPP ( u 3 ǫ 2 ) PPP ( u 2 ǫ 2 ) PPP ( uǫ 2 ) PPP ( ǫ 2 ) With L the longest up-right path in this cylindric geometry, in distribution, Schensted’s theorem states that λ 1 = L + κ 1 where κ is a uniform partition Prob ( κ ) ∝ u | κ | independent of everything else.
A word on the finite temperature Airy kernel Ai α ◮ introduced by Johansson (Joh07) ◮ also appearing as the KPZ crossover kernel: SasSpo10 and AmiCorQua11; in random directed polymers BorCorFer11; cylindric OU processes LeDMajSch15 ◮ interpolates between the Airy kernel and a diagonal exponential kernel: α →∞ Ai α ( x , y ) = Ai 2 ( x , y ) , lim � x 1 α − 1 2 α log(4 πα 3 ) , y α − 1 � α Ai α 2 α log(4 πα 3 ) = e − x δ x , y lim α → 0+ ◮ with F α ( s ) , F TW ( s ), and G ( s ) the Fredholm determinants on ( s , ∞ ) of Ai α , Ai 2 and e − x δ x , y , (Joh07) α →∞ F α ( s ) = F TW ( s ) , lim � s α − 1 � = G ( s ) = e − e − s α → 0+ F α 2 α log(4 πα 3 ) lim
Direct limit to Tracy–Widom Theorem (B/Bouttier 2019) Let u → 1 and ǫ → ∞ in such a way that ǫ (1 − u ) 2 → ∞ . Then we have � λ 1 − 2 M � ǫ Prob ≤ s → F TW ( s ) , M := 1 − u . M 1 / 3
Direct limit to Gumbel Theorem (B/Bouttier 2019) Set u = e − r and assume that r → 0+ and ǫ r 2 → 0+ (with ǫ possibly remaining finite). Then: � r λ 1 − ln I 0 (2 ǫ + ǫ r ) � → e − e − s Prob ≤ s r � π 1 − π e x cos φ d φ is the modified Bessel function of the first kind and order where I 0 ( x ) := 2 π zero.
Thank you!
Last passage percolation Character identities and LPP Duality between determinants and Pfaffians Transition between characters of classical groups, decomposition of Gelfand-Tsetlin paterns, and last passage percolation (joint work with Nikos Zygouras) Elia Bisi University College Dublin Integrability and Randomness in Mathematical Physics and Geometry Marseille, 9 April 2019 1 / 4
Last passage percolation Character identities and LPP Duality between determinants and Pfaffians Last passage percolation (LPP) � L ( 2 n , 2 n ) : = max W i , j π ∈ Π 2 n , 2 n ( i , j ) ∈ π Π 2 n , 2 n is the set of directed paths in { 1 ,..., 2 n } 2 starting from ( 1 , 1 ) and ending at ( 2 n , 2 n ) ; { W i , j } 1 ≤ i , j ≤ 2 n is a field of independent geometric random variables with various symmetries (1 , 1) (1 , 1) (1 , 1) (2 n, 2 n ) (2 n, 2 n ) (2 n, 2 n ) Antidiagonal symmetry Diagonal symmetry Double symmetry 2 / 4
Last passage percolation Character identities and LPP Duality between determinants and Pfaffians Character identities and LPP � 2 n � i = 1 ( µ i mod 2 ) · s ( 2 n ) � � L β ( 2 n , 2 n ) ≤ 2 u β ( p 1 ,..., p 2 n ) P ∝ µ µ ⊆ ( 2 u ) ( 2 n ) u 2 n � s CB p i u ( 2 n ) ( p 1 ,..., p 2 n ; β ) = i = 1 u � 2 n � s CB λ ( p 1 ,..., p n ; β ) · s CB p i λ ( p n + 1 ,..., p 2 n ; β ) = i = 1 λ ⊆ u ( n ) (1 , 1) s ( 2 n ) is a classical Schur polynomial; µ s CB λ is a Schur polynomial that interpolates between symplectic and odd orthogonal characters. (2 n, 2 n ) 3 / 4
Last passage percolation Character identities and LPP Duality between determinants and Pfaffians Duality between determinants and Pfaffians Baik-Rains’ formulas and ours show a duality between Pfaffians and determinants, for finite N . Fredholm Pfaffian and Fredholm determinantal expressions of the limiting distribution functions, as N → ∞ . E.g., we obtain: Sasamoto’s Fredholm determinant for the GOE Tracy-Widom distribution in the case of antidiagonal symmetry : F 1 ( s ) = det ( I − B s ) Ferrari-Spohn’s Fredholm determinant for the GSE Tracy-Widom distribution in the case of diagonal symmetry : � � F 4 ( s ) = 1 det ( I − B √ 2 s ) + det ( I + B √ 2 s ) 2 with the kernel being B s ( x , y ) : = Ai ( x + y + s ) on L 2 ( [0 , ∞ )) . 4 / 4
Painlev´ e II τ -function as a Fredholm determinant Harini Desiraju SISSA, Trieste
Introduction • Question: Can the τ -function of Painlev´ e II be expressed as a Fredholm determinant? 1
Introduction • Question: Can the τ -function of Painlev´ e II be expressed as a Fredholm determinant? • Painlev´ e II q ss = sq − 2 q 3 (1) 1
Introduction • Question: Can the τ -function of Painlev´ e II be expressed as a Fredholm determinant? • Painlev´ e II q ss = sq − 2 q 3 (1) • The τ -function of Painlev´ e II is related to its transcendent d 2 ds 2 ln τ [ s ] = − q 2 ( s ) (2) 1
Introduction • Question: Can the τ -function of Painlev´ e II be expressed as a Fredholm determinant? • Painlev´ e II q ss = sq − 2 q 3 (1) • The τ -function of Painlev´ e II is related to its transcendent d 2 ds 2 ln τ [ s ] = − q 2 ( s ) (2) • What is known? 1
Introduction • Question: Can the τ -function of Painlev´ e II be expressed as a Fredholm determinant? • Painlev´ e II q ss = sq − 2 q 3 (1) • The τ -function of Painlev´ e II is related to its transcendent d 2 ds 2 ln τ [ s ] = − q 2 ( s ) (2) • What is known? • Ablowitz-Segur family is a special solution of PII q ( s ) ≈ κ Ai ( s ); κ ∈ C ; s → ∞ (3) 1
Introduction • Question: Can the τ -function of Painlev´ e II be expressed as a Fredholm determinant? • Painlev´ e II q ss = sq − 2 q 3 (1) • The τ -function of Painlev´ e II is related to its transcendent d 2 ds 2 ln τ [ s ] = − q 2 ( s ) (2) • What is known? • Ablowitz-Segur family is a special solution of PII q ( s ) ≈ κ Ai ( s ); κ ∈ C ; s → ∞ (3) • It is a known result that the τ -function in this case is the determinant of the Airy Kernel. τ [ s ] = det[ I − κ 2 K Ai ] | [ s , ∞ )] (4) 1
General Painlev´ e II using IIKS construction • The Riemann Hilbert problem of Painlev´ e II, after some transformations, can be reduced to the following RHP on i R Γ( z ) = 1 + O ( z − 1 ) Γ + ( z ) = Γ − ( z ) J ( z ); z → ∞ (5) as χ 1 χ 2 i R χ 3 χ 4 2
General Painlev´ e II using IIKS construction • The Riemann Hilbert problem of Painlev´ e II, after some transformations, can be reduced to the following RHP on i R Γ( z ) = 1 + O ( z − 1 ) Γ + ( z ) = Γ − ( z ) J ( z ); z → ∞ (5) as χ 1 χ 2 i R χ 3 χ 4 � a ( z ) b ( z ) � = 1 − 2 π if ( z ) g T ( z ) • Using χ i , the jump function is J ( z ) = c ( z ) d ( z ) 2
General Painlev´ e II using IIKS construction • The Riemann Hilbert problem of Painlev´ e II, after some transformations, can be reduced to the following RHP on i R Γ( z ) = 1 + O ( z − 1 ) Γ + ( z ) = Γ − ( z ) J ( z ); z → ∞ (5) as χ 1 χ 2 i R χ 3 χ 4 � a ( z ) b ( z ) � = 1 − 2 π if ( z ) g T ( z ) • Using χ i , the jump function is J ( z ) = c ( z ) d ( z ) • with χ 2 ( z ) + ( b ( z ) − 1) � � � � χ 4 ( z ) 1 χ 1 ( z ) + χ 3 ( z ) a ( z ) f ( z ) = ; g ( z ) = (1+ c ( z ) − a ( z )) 2 π i χ 2 ( z ) + χ 4 ( z ) χ 1 ( z ) + ( a ( z ) − 1) χ 3 ( z ) a ( z ) a ( z ) , b ( z ) , c ( z ) , d ( z ) are given in terms of parabolic cylinder functions. 2
Results • The integrable kernel on L 2 ( i R ) is given by K ( z , w ) = f T ( z ) g ( w ) (6) 2 π i ( z − w ) 3
Results • The integrable kernel on L 2 ( i R ) is given by K ( z , w ) = f T ( z ) g ( w ) (6) 2 π i ( z − w ) • τ -function: τ [ s ] = det( 1 − K ) (7) 3
Results • The integrable kernel on L 2 ( i R ) is given by K ( z , w ) = f T ( z ) g ( w ) (6) 2 π i ( z − w ) • τ -function: τ [ s ] = det( 1 − K ) (7) • τ [ s ] is related to the JMU τ -function as 3 + ν 2 � 2 i ν � ∂ s ln τ [ s ] = ∂ s ln τ JMU + + A ( ν ) (8) s where ν = − 1 2 π i ln(1 − s 1 s 3 ) and s 1 , s 3 are Stokes’ parameters and s is the PII parameter and A ( ν ) is a non-vanishing depending only on ν . 3
References Fokas, A.S., Its, A.R., Kapaev, A.A., Kapaev, A.I., Novokshenov, V.Y. and Novokshenov, V.I., 2006. Painlev´ e transcendents: the Riemann-Hilbert approach (No. 128). American Mathematical Soc. Bertola, M., 2017. The Malgrange form and Fredholm determinants. arXiv preprint arXiv:1703.00046. Its, A.R., Izergin, A.G., Korepin, V.E. and Slavnov, N.A., 1990. Differential equations for quantum correlation functions. International Journal of Modern Physics B, 4(05), pp.1003-1037. Cafasso, M., Gavrylenko, P. and Lisovyy, O., 2017. Tau functions as Widom constants . arXiv preprint arXiv:1712.08546. Bothner, T. and Its, A., 2012. Asymptotics of a Fredholm determinant involving the second Painlev´ e transcendent. arXiv preprint arXiv:1209.5415. 4
Extreme gap problems in random matrix theory Renjie Feng BIMCR, Peking University Renjie Feng (BICMR) 1 / 10
Previous results I: smallest gaps for CUE Let e i θ 1 , · · · , e i θ n be n eigenvalues of CUE, consider n � χ n = δ ( n 4 / 3 ( θ i +1 − θ i ) ,θ i ) . i =1 Theorem (Vinson, Soshnikov, Ben Arous-Bourgade) χ n tends to a Poisson process χ with intensity � 1 � � �� du � u 2 du E χ ( A × I ) = . 24 π 2 π A I The k th smallest gap has limiting density 3 ( k − 1)! x 3 k − 1 e − x 3 . Renjie Feng (BICMR) 2 / 10
Previous results II: smallest gaps for GUE For GUE n � χ n = δ ( n 3 ( λ i +1 − λ i ) ,λ i ) 1 | λ i | < 2 − η 4 i =1 Theorem (Ben Arous-Bourgade, AOP 2013) χ n tends to a Poisson process χ with intensity 1 � � u 2 du )( (4 − x 2 ) 2 dx ) , E χ ( A × I ) = ( 48 π 2 A I where A ⊂ R + and I ⊂ ( − 2 + η, 2 − η ). ( k − 1)! x 3 k − 1 e − x 3 , same as 3 The k th smallest gap has the limiting density CUE. Renjie Feng (BICMR) 3 / 10
New results I: smallest gaps for C β E When β is an positive integer, consider n � χ n = δ β +2 β +1 ( θ i +1 − θ i ) ,θ i ) ( n i =1 Theorem [F.-Wei] χ n tends to a Poisson point process χ with intensity E χ ( A × I ) = A β | I | � u β du , 2 π A where A β = (2 π ) − 1 ( β/ 2) β (Γ( β/ 2+1)) 3 Γ(3 β/ 2+1)Γ( β +1) . For COE, CUE and CSE, A 1 = 1 1 1 24 , A 2 = 24 π, A 4 = 270 π. Renjie Feng (BICMR) 4 / 10
New results II: smallest gaps for GOE For GOE n − 1 χ ( n ) = � δ n 3 / 2 ( λ ( i +1) − λ ( i ) ) i =1 Theorem [F.-Tian-Wei] χ ( n ) converges to a Poisson point process χ with intensity E χ ( A ) = 1 � udu . 4 A the limiting density of the k th smallest gap is 2 ( k − 1)! x 2 k − 1 e − x 2 , same as COE. Conjecture: C β E and G β E share the same smallest gaps. Renjie Feng (BICMR) 5 / 10
Previous III: order of largest gaps For CUE and interior of GUE, m k is the k th largest gap, Theorem (Ben Arous-Bourgade, AOP 2013) For any p > 0 and l n = n o (1) , one has n L p m l n × √ → 1 . 32 ln n Renjie Feng (BICMR) 6 / 10
New results III: fluctuation of largest gaps Theorem (F.-Wei) Let’s denote m k as the k -th largest gap of CUE, and 1 1 τ n 2 ( nm k − (32 ln n ) 2 ) / 4 − (3 / 8) ln(2 ln n ) , k = (2 ln n ) then { τ n k } tends to a Poisson process and τ n k has the limit of the Gumbel distribution, e k ( c 1 − x ) ( k − 1)! e − e c 1 − x . Here, c 1 = 1 12 ln 2 + 3 ζ ′ ( − 1) + ln π 2 . Renjie Feng (BICMR) 7 / 10
New results III: fluctuation of largest gaps Theorem (F.-Wei) √ 4 − x 2 and Let’s denote m ∗ k as the k -th largest gap of GUE, S ( I ) = inf I 1 1 τ ∗ 2 ( nS ( I ) m ∗ 2 ) / 4 + (5 / 8) ln(2 ln n ) , k = (2 ln n ) k − (32 ln n ) { τ ∗ k } tends to a Poisson process and has the limit of the Gumbel distribution, e k ( c 2 − x ) ( k − 1)! e − e c 2 − x . Here, c 2 = 1 12 ln 2 + 3 ζ ′ ( − 1) + M 0 ( I ) depending on I , where M 0 ( I ) = (3 / 2) ln(4 − a 2 ) − ln(4 | a | ) if a + b < 0 , M 0 ( I ) = (3 / 2) ln(4 − b 2 ) − ln(4 | b | ) if a + b > 0, M 0 ( I ) = (3 / 2) ln(4 − a 2 ) − ln(2 | a | ) if a + b = 0 . Renjie Feng (BICMR) 8 / 10
Extreme gaps IV: universality of extreme gaps Recently, our results are generalized for Hermitian/symmetric Wigner matrices with mild assumptions. P. Bourgade, Extreme gaps between eigenvalues of Wigner matrices, arXiv:1812.10376. B. Landon, P. Lopatto, J. Marcinek, Comparison theorem for some extremal eigenvalue statistics, arXiv:1812.10022. Renjie Feng (BICMR) 9 / 10
References Large gaps of CUE and GUE, arXiv:1807.02149. Small gaps of circular beta-ensemble, arXiv:1806.01555 Small gaps of GOE, arXiv:1901.01567. Renjie Feng (BICMR) 10 / 10
MATRIX MODELS FOR CLASSICAL GROUPS AND TOEPLITZ+HANKEL MINORS WITH APPLICATIONS TO CHERN-SIMONS THEORY AND FERMIONIC MODELS David García-García Joint work with Miguel Tierz (arXiv:1901.08922)
MINORS OF TOEPLITZ+HANKEL MATRICES d 1 d 2 d 7 d 3 d 6 d 4 d 10 d 2 d 9 d 8 d 1 d 0 d 7 d 1 d 6 d 2 d 5 d 3 d 9 d 3 d 8 d 9 d 2 . . . . . . . . . . . d 0 . . . . . . d 11 d 1 d 10 d 8 d 7 U N d 1 d 6 d 1 d 5 d 3 d 4 d 2 d 3 d 2 d 7 d 0 N N det N 1 f M dM d 5 d 4 d 1 d 4 d 6 d 0 d 5 d 1 d 3 d 2 d 8 d 4 d 7 d 3 d 6 d 2 d 5 d 1 d 4 d 0 . ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N | ∆( e i θ ) | 2 ∏ N ! 2 π k = 1
MINORS OF TOEPLITZ+HANKEL MATRICES d 1 d 2 d 7 d 3 d 6 d 4 d 10 d 2 d 9 d 8 d 1 d 0 d 7 d 1 d 6 d 2 d 5 d 3 d 9 d 3 d 8 d 9 d 2 . . . . . . . . . . . d 0 . . . . . . d 11 d 1 d 10 d 8 d 7 . d 3 d 6 d 1 d 5 d 3 d 4 d 2 d 1 d 7 d 2 d 0 N N det N 1 d 5 d 4 d 1 d 4 d 6 d 0 d 5 d 3 d 4 d 2 d 8 d 1 d 7 d 3 d 6 d 2 d 5 d 1 d 0 d 4 ∫ f ( M ) dM = U ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N | ∆( e i θ ) | 2 ∏ 2 π = N ! k = 1
MINORS OF TOEPLITZ+HANKEL MATRICES d 5 d 3 d 7 d 4 d 8 d 2 d 4 d 1 d 0 d 2 d 6 d 1 d 7 d 2 d 8 d 3 d 9 d 6 d 5 d 5 d 3 N d 0 d 2 d 1 d 3 d 2 d 4 d 5 d 1 d 4 d 6 d 5 d 7 d 1 d 3 d 0 d 4 d 3 d 2 det . . . . . . . . . d 11 . . . . . . . . d 1 d 6 d 10 d 1 d 7 d 0 d 8 d 1 d 9 d 2 d 4 d 10 d 6 d 3 d 7 d 2 d 8 d 1 d 9 d 0 N . d 3 d 0 . . . . d 0 d 1 d 2 d 4 d 1 . d 2 d 3 det d 0 d 0 d 1 d 2 d 0 d 1 . N . . . . . . 1 . . . . . . ∫ f ( M ) dM = U ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N | ∆( e i θ ) | 2 ∏ 2 π = N ! k = 1 · · · d − 1 d − 2 d − 3 d − 4 d − 5 · · · d − 1 d − 2 d − 3 d − 4 · · · d − 1 d − 2 d − 3 · · · d − 1 d − 2 N × N · · · d − 1
MINORS OF TOEPLITZ+HANKEL MATRICES d 4 d 2 d 6 d 3 d 7 d 4 d 8 d 2 d 1 d 1 d 5 d 0 d 6 d 1 d 7 d 2 d 8 d 3 d 5 d 4 d 3 d 2 det N N d 0 d 2 d 1 d 3 d 4 d 0 d 3 d 5 d 4 d 6 d 5 d 7 d 1 d 3 d 9 d 5 . . . . . . . . . . d 1 . . . . . . . . d 11 d 10 d 2 d 2 d 6 d 1 d 7 d 0 d 8 d 1 d 9 d 10 d 0 d 4 d 6 d 3 d 7 d 2 d 8 d 1 d 9 . . . d 0 d 2 d 3 3 d 2 d 1 d . d 1 d 2 2 d d 1 d 0 5 d 4 d 3 d 2 d 1 d d 0 det N 1 d 1 d 0 d 4 . . . . . . . . . . . . d 0 . . d 3 d 2 d 1 ∫ f ( M ) dM = U ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N | ∆( e i θ ) | 2 ∏ 2 π = N ! k = 1 · · · d − 1 d − 3 d − 4 · · · d − 1 d − 2 N × N · · · d − 1
MINORS OF TOEPLITZ+HANKEL MATRICES d 4 d 2 d 6 d 3 d 7 d 4 d 8 d 2 d 1 d 1 d 5 d 0 d 6 d 1 d 7 d 2 d 8 d 3 d 5 d 4 d 3 d 2 det N N d 0 d 2 d 1 d 3 d 4 d 0 d 3 d 5 d 4 d 6 d 5 d 7 d 1 d 3 d 9 d 5 . . . . . . . . . . d 1 . . . . . . . . d 11 d 10 d 2 d 2 d 6 d 1 d 7 d 0 d 8 d 1 d 9 d 10 d 0 d 4 d 6 d 3 d 7 d 2 d 8 d 1 d 9 . . . d 0 d 2 d 3 3 d 2 d 1 d . d 1 d 2 2 d d 1 d 0 5 d 4 d 3 d 2 d 1 d d 0 det N 1 d 1 d 0 d 4 . . . . . . . . . . . . d 0 . . d 3 d 2 d 1 ∫ s λ ( M − 1 ) s µ ( M ) f ( M ) dM = U ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N s λ ( e − i θ ) s µ ( e i θ ) | ∆( e i θ ) | 2 ∏ 2 π = N ! k = 1 · · · d − 1 d − 3 d − 4 · · · d − 1 d − 2 N × N · · · d − 1
MINORS OF TOEPLITZ+HANKEL MATRICES d 8 d 2 d 7 d 3 d 6 d 4 d 10 d 2 d 9 d 1 d 0 d 1 d 7 d 1 d 6 d 2 d 5 d 3 d 9 d 3 d 8 d 8 d 9 d 7 . . . . . . . . . . . d 0 . . . . . . d 11 d 1 d 10 d 2 d 1 . d 3 d 6 d 4 d 6 d 3 d 4 d 2 d 1 d 7 d 2 d 0 N N det N 1 d 5 d 5 d 1 d 4 d 0 d 5 d 3 d 4 d 2 d 8 d 1 d 7 d 3 d 6 d 2 d 5 d 1 d 0 d 4 ∫ f ( M ) dM = ( G ( N ) = Sp ( 2 N ) , O ( 2 N ) , O ( 2 N + 1 )) G ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N | ∆ G ( N ) ( e i θ ) | 2 ∏ 2 π = N ! k = 1
MINORS OF TOEPLITZ+HANKEL MATRICES . . . . . . . . . . . . . . . . . . N 1 det ∫ f ( M ) dM = ( G ( N ) = Sp ( 2 N ) , O ( 2 N ) , O ( 2 N + 1 )) G ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N | ∆ G ( N ) ( e i θ ) | 2 ∏ 2 π = N ! k = 1 d 0 − d 2 d 1 − d 3 d 2 − d 4 d 3 − d 5 d 4 − d 6 d 5 − d 7 · · · d 1 − d 3 d 0 − d 4 d 1 − d 5 d 2 − d 6 d 3 − d 7 d 4 − d 8 · · · d 2 − d 4 d 1 − d 5 d 0 − d 6 d 1 − d 7 d 2 − d 8 d 3 − d 9 · · · d 3 − d 5 d 2 − d 6 d 1 − d 7 d 0 − d 8 d 1 − d 9 d 2 − d 10 · · · N × N d 4 − d 6 d 3 − d 7 d 2 − d 8 d 1 − d 9 d 0 − d 10 d 1 − d 11 · · ·
MINORS OF TOEPLITZ+HANKEL MATRICES d 2 d 8 d 0 d 6 d 2 d 9 d 3 d 8 d 7 d 7 d 1 d 6 d 0 d 1 d 4 d 2 d 6 d 2 d 3 d 1 d 0 . . . . . . . . . . d 9 . . . . . . . . d 4 d 5 N d 3 1 det d 0 d 2 d 3 d 2 d 4 d 1 d 5 d 4 d 5 d 7 d 6 ∫ χ λ G ( N ) ( M − 1 ) χ µ G ( N ) ( M ) f ( M ) dM = ( G ( N ) = Sp ( 2 N ) , O ( 2 N ) , O ( 2 N + 1 )) G ( N ) ∫ f ( e i θ k ) d θ k [ 0 , 2 π ] N χ λ G ( N ) ( e − i θ ) χ µ G ( N ) ( e i θ ) | ∆ G ( N ) ( e i θ ) | 2 ∏ 2 π = N ! k = 1 d 1 − d 3 d 1 − d 5 d 3 − d 7 d 4 − d 8 · · · d 3 − d 5 d 1 − d 7 d 1 − d 9 d 2 − d 10 · · · N × N d 4 − d 6 d 2 − d 8 d 0 − d 10 d 1 − d 11 · · ·
SOME RESULTS AND APPLICATIONS · Factorizations · Chern-Simons theory N 2 N · Expansions in terms of Toeplitz minors Partition function ∫ ∫ ∫ f ( U ) dU = f ( U ) dU f ( − U ) dU , U ( 2 N ) O ( 2 N + 1 ) O ( 2 N + 1 ) ∫ ∫ ∫ f ( U ) dU = f ( U ) dU f ( U ) dU . U ( 2 N + 1 ) Sp ( 2 N ) O ( 2 N + 2 ) ∑ ( − 1 ) ( | λ | + | µ | ) / 2 D λ,µ det ( d j − k − d j + k ) N j , k = 1 = 1 ( f ) . λ,µ ∈ R ( N ) ∫ Θ( U ) dU G ( N )
SOME RESULTS AND APPLICATIONS · Factorizations · Chern-Simons theory N 2 N · Expansions in terms of Toeplitz minors Wilson loop ∫ ∫ ∫ f ( U ) dU = f ( U ) dU f ( − U ) dU , U ( 2 N ) O ( 2 N + 1 ) O ( 2 N + 1 ) ∫ ∫ ∫ f ( U ) dU = f ( U ) dU f ( U ) dU . U ( 2 N + 1 ) Sp ( 2 N ) O ( 2 N + 2 ) ∑ ( − 1 ) ( | λ | + | µ | ) / 2 D λ,µ det ( d j − k − d j + k ) N j , k = 1 = 1 ( f ) . λ,µ ∈ R ( N ) ∫ χ µ G ( N ) ( U )Θ( U ) dU G ( N )
SOME RESULTS AND APPLICATIONS · Factorizations · Chern-Simons theory N 2 N · Expansions in terms of Toeplitz minors Hopf link ∫ ∫ ∫ f ( U ) dU = f ( U ) dU f ( − U ) dU , U ( 2 N ) O ( 2 N + 1 ) O ( 2 N + 1 ) ∫ ∫ ∫ f ( U ) dU = f ( U ) dU f ( U ) dU . U ( 2 N + 1 ) Sp ( 2 N ) O ( 2 N + 2 ) ∑ ( − 1 ) ( | λ | + | µ | ) / 2 D λ,µ det ( d j − k − d j + k ) N j , k = 1 = 1 ( f ) . λ,µ ∈ R ( N ) ∫ χ λ G ( N ) ( U − 1 ) χ µ G ( N ) ( U )Θ( U ) dU G ( N )
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Semigroups for One-Dimensional Schr¨ odinger Operators with Multiplicative White Noise 1 Pierre Yves Gaudreau Lamarre Princeton University 1 Based on a paper of the same name; arXiv:1902.05047. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
Let ξ be a Gaussian white noise on R d , and V : R d → R be a deterministic function. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
Let ξ be a Gaussian white noise on R d , and V : R d → R be a deterministic function. Consider the random Schr¨ odinger operator ˆ � − 1 � H := 2 ∆ + V + ξ. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
Let ξ be a Gaussian white noise on R d , and V : R d → R be a deterministic function. Consider the random Schr¨ odinger operator ˆ � − 1 � H := 2 ∆ + V + ξ. Problem. Develop a semigroup theory for ˆ H , i.e., � H : t > 0 � e − t ˆ Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
1 SPDEs: u ( t, x ) := e − t ˆ H u 0 ( x ) solves � 1 � ∂ t u = 2 ∆ − V u + ξu, u (0 , x ) = u 0 ( x ) . Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
1 SPDEs: u ( t, x ) := e − t ˆ H u 0 ( x ) solves � 1 � ∂ t u = 2 ∆ − V u + ξu, u (0 , x ) = u 0 ( x ) . 2 Spectral Analysis of SPDEs: ∞ e − t ˆ e − tλ k ( ˆ H ) � ψ k ( ˆ H ) , u 0 � ψ k ( ˆ � H u 0 = H ) . k =1 Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
1 SPDEs: u ( t, x ) := e − t ˆ H u 0 ( x ) solves � 1 � ∂ t u = 2 ∆ − V u + ξu, u (0 , x ) = u 0 ( x ) . 2 Spectral Analysis of SPDEs: ∞ e − t ˆ e − tλ k ( ˆ H ) � ψ k ( ˆ H ) , u 0 � ψ k ( ˆ � H u 0 = H ) . k =1 3 Feynman-Kac formula: e − t ˆ H f ( x ) � t � � � �� = E x � � � � � exp − V B ( s ) + ξ B ( s ) d s f B ( t ) . 0 Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
ξ cannot be defined pointwise. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
ξ cannot be defined pointwise. It is thus nontrivial to define ˆ − 1 � � Hf = 2 ∆ + V f + ξf ; Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
ξ cannot be defined pointwise. It is thus nontrivial to define ˆ − 1 � � Hf = 2 ∆ + V f + ξf ; � t � � � �� E x � � � � � exp − V B ( s ) + ξ B ( s ) d s f B ( t ) 0 Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
At my poster: Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
At my poster: 1 Show how these technical obstacles can be overcome in one dimension. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
At my poster: 1 Show how these technical obstacles can be overcome in one dimension. 2 Discuss applications in random matrix theory and SPDEs with multiplicative white noise. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
At my poster: 1 Show how these technical obstacles can be overcome in one dimension. 2 Discuss applications in random matrix theory and SPDEs with multiplicative white noise. 3 Discuss partial results in higher dimensions, and connection to regularity structures/paracontrolled calculus/renormalization of SPDEs. Pierre Yves Gaudreau Lamarre Semigroups for 1D Operators with Noise
The longest increasing subsequence problem for correlated random variables J. Ricardo G. Mendonça LPTMS, CNRS, Université Paris-Sud, Université Paris Saclay, 91405 Orsay, France Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, SP, Brasil L’Intégrabilité et l’Aléatoire en Physique Mathématique et en Géométrie CIRM, Marseille Luminy, France, 8–12 April 2019
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