A tale of Pfaffian persistence tails told by a Bonnet-Painlev´ e VI transcendent Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 ID, Pfaffian Persistence, Universal Bonnet-Painlev´ e VI Probability Distributions, and Ising model Criticalities in 1+1 dimensions , arXiv:1810.06957 (under rev. for J. Stat. Phys.) Robert Conte & ID, Persistence, Painlev´ e VI, Chazy C.V, and Bonnet Surfaces , in prep. ( < 2020) — for a genuine (applied) maths journal ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Appetizer: What is a Bonnet-B P VI surface ? Let P L ( x ) = � L k =0 a k x k the usual Kac polynomial with real Gaussian random coeffs, and N = N ( L ) the number of its real roots on [0 , 1]. Then for L ≫ 1 1 E [ N ] ∼ 2 π ln L (Kac 1943, Thm) What about the full distribution: E [ m N ], with 0 < m < 1 ? My claim: ∃ scaling function of T ≡ ln L ∈ (0 , + ∞ ): � � T �� � � − 1 E [ m N ] → exp H ′ ( ℓ ) H ( ℓ ) + 2 0 H = H ( T ; m ) the (extrinsic) mean curvature of the above, also the 2 π , H ′ (0) = H (0) 2 of sole ր regular solution on R + with H (0) = 1 − m 2 � � 2 � � 1 � 2 � H ′′ � 2 � H ′ + H 2 = 2 H ′ + coth T 1 − H + coth T H ′ 2 ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Appetizer (cont’d) and having a finite limit for large times T : T → + ∞ H ( T ; m ) = 1 θ ( m ) = lim 2 [ κ 1 ( m ) + κ 2 ( m )] (1) �� 2 �� 2 � � m �� 2 � 2 � 1 1 √ − √ = π arccos π arccos (2) 2 2 2 This recovers — independently and by completely different methods — a result just obtained (and before . . . ) by Poplavsky & Schehr’: E [ m N ( L )] ∝ L − θ ( m ) / 2 ∝ e − θ ( m ) T / 2 , T ” = ” ln L ≫ 1 Exact persistence exponent for the 2 d-diffusion equation and related Kac polynomials , Phys. Rev. Lett. 2018 (arXiv:1806.11275) Both answering a famous question by Dembo et al about Random polynomials having few or no real zeros (J. AMS 2002) θ (0) = 3 m → 0 + E [ m N ( L ) ] ∝ L − θ (0) / 2 , lim 16 ≡ Gauss ( intrinsic ) curvature 2 ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Appetizer (bonuses) Yet here bonus: H is a tau-function for a P VI with monodromy exponents (up to perm./signs) or parameters � 1 � � 1 � 2 , 1 8 , − 1 8 , 0 , 1 { ϑ ∞ , ϑ 0 , ϑ 1 , ϑ s } = 2 , 0 , 0 , { α, β, γ, δ } = 2 Universal ` a la Tracy-Widom: it appears in four different model systems of interest for nonequilibrium statistical physics Halfway (through quadratic+ Okamoto transformations) between some other famous P VI : Picard/Hitchin, & Manin ⇒ related to Jimbo-Miwa’s characterization of the diagonal correlations of the planar equilibrium Ising model at all temperatures � sinh 2 E � 2 θ (0) = 3 16 = η + β = 1 / 4 + 1 / 8 , tanh ( T / 2) = sinh 2 E ∗ 2 2 2 . The reason for all this: Pfaffian structure with an integrable kernel, 2 π sech ( x − y ) / 2, on L 2 ( − T / 2 , T / 2) 1 the sech-kernel K ( x , y ) = ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Introduction: First-passage properties of a random process What is the chance for a fluctuating quantity Y to have always remained up to a certain time above a given level (say � Y � ), or to first cross it a certain instant ? Time-honored and basic subject of (applied) probability Usual playground: { Y ( T ) } T Gaussian stationary process, thus A ( T 2 − T 1 ) = � Y ( T 1 ) Y ( T 2 ) � determines everything a priori P 0 ( T ) = No-crossing proba. at zero level (= � Y � ), up to (fixed) T : P 0 ( T ) ∝ e − θ T , θ = decay rate − d P 0 ( T ) / d T = first-passage proba. at time T Simple pb to state but extremely hard to solve unless for Markovian (memoryless) processes . . . The latter have necessarily A ( T ) = e − θ T ∀ T (Slepian’s theorem), hence Y ≡ rescaled Brownian: P 0 ( T ) = (2 /π ) arcsin A ( T ) ∝ e − θ T ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Modern incarnation: persistence in phase-ordering systems How a local degree of freedom can maintain its initial orientation as domains of globally aligned spins grow as L ( t ) ∝ t 1 / z ? Simplest situation: quench ± Ising spins from infinite to zero temperature. Introduce somewhat natural geometric definition: p 0 ( t ) = fraction of spins which have never flipped up to time t : − d p 0 ( t ) / d t = first-passage probability. of a domain wall at a particular location in space. For large times algebraic decay p 0 ( t ) ∝ t − θ , θ = persistence exponent t GF .J a l. P I I ^- FIc. 2.1 - Paysage des d,omaines d,ans le modèle d,'Ising 2d, éaoluant selon la d,ynamique de Glauber àT -- 0, pour d,es temps t - 2b6,r024,40g6 d,ans un système d'e N : (512)2 sites (aaec des condit,ions aur li.mit,es péri,od,íques). ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Two early climaxes in the physical literature circa 1995 Simple diffusion equation ∂ t φ ( r , t ) = ∇ 2 r φ ( r , t ), with φ ( r , 0) = white noise. A popular model of phase ordering because φ ( r , t ) Gaussian. Local ”spin” variable (at r = 0 say) sgn [ X ( t )], X ( t ) = φ (0 , t ) Yet non-trivial ˜ θ ( d ) in all space dimensions d ! (Majumdar, Sire, Bray, Cornell & Derrida, Hakim, Zeitak, PRLs ’95) ”Simply” because non-Markovian correlator for the associated process Y ( T ) = X ( e T ) / [ � X 2 ( e T ) � ] 1 / 2 (normalized and rendered stationary on the logarithmic timescale T = ln t ) In particular in d = 2, ˜ θ (2) = 0 . 1875(10) (num.) with a correlator A ( T ) = � Y (0) Y ( T ) � = sech ( T / 2) ( sech = 1 / cosh) Later realized (Dembo et al., Schehr-Majumdar, Forrester . . . ) that the very same Gaussian { Y ( T ) } T also describes the number of real roots of random Kac’s polynomials or the eigenvalues of truncated random orthogonal matrices, both Pfaffian point processes ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Second climax: Derrida, Hakim, Pasquier’s tour-de-force An exact expression for the persistence proba. p Potts ( t 1 , t 2 ; q ) that a 0 q -state Potts spin on a 1 d chain with zero-temperature Glauber dynamics has not flipped between (arbitrary) times t 1 , t 2 θ ( q ) with ( t 1 , t 2 ; q ) ∝ ( t 2 / t 1 ) − ˆ After tremendous technicalities, p Potts 0 � � 2 − q �� 2 θ ( q ) = − 1 8 + 2 ˆ ⇒ ˆ arccos √ = θ (2) = 3 / 8 ( Ising spins ) π 2 2 q Their crucial insight: the pers. proba. for the particular spin located at the origin of a semi-infinite chain is determined by the Pfaffian formed by the no-meeting proba. c ( s , t ) between two random walkers � s � [ p ( x ; s ) p ( y ; t ) − p ( x ; t ) p ( y ; s )] ≈ 1 − 4 c ( s , t ) = π arctan t 0 ≤ x ≤ y ( t 1 , t 2 ; q )] 1 / 2 ∝ ( t 2 / t 1 ) − ˆ Enough since p SemiP ( t 1 , t 2 ; q ) = [ p Potts θ ( q ) / 2 0 0 ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
Yet . . . Verbatim from the conclusions of DHP (J.Stat.Phys.’96): This probably means that there are simpler ways of rewriting our expression (for p SemiP ( t 1 , t 2 ; q ) ) where all the cases can be treated in 0 the same manner. Unfortunately, we did not find these simpler expressions. Puzzling numerical proximity θ ( q = 2) / 2 = 1 ˆ ˜ 2(3 / 8) vs . θ ( d = 2) = 0 . 1875(10) But how these two model systems, apparently so dissimilar, and that do not even live in an ambient space with the same physical dimension, could possibly be related at the level of a quantity so sensitive to details as the persistence exponent? :-( Just proved by M. Poplavskyi & G. Schehr! Exact persistence exponent for the 2 d-diffusion equation and related Kac polynomials , arXiv:1806.11275 (PRL in press) sgn X Diff2d ( t ) ≡ S SemiIsing ( t ) ( as processes ) 0 ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
What else ? Different and More (if possible. . . ) I had (vague) indications of a Painlev´ e VI lurking in the background and my hope was that this could allow to rederive PS’s result ”somehow” ID, Pfaffian Persistence, Universal Bonnet-Painl Ivan Dornic (CEA Saclay & Sorbonne Univ., Condensed Matter Labs) CIRM — April 12, 2019 Persistence, Bonnet P VI , & Ising / 33
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