telegraph equation from the six vertex model
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Telegraph equation from the six-vertex model Vadim Gorin MIT - PowerPoint PPT Presentation

Dec 30, 2023 •377 likes •818 views

Telegraph equation from the six-vertex model Vadim Gorin MIT (Cambridge) and IITP (Moscow) July 2018 Telegraph equation Resistance R Inductance L Capacitance C Conductance G Wiki/Pixabay

  1. Telegraph equation from the six-vertex model Vadim Gorin MIT (Cambridge) and IITP (Moscow) July 2018

  2. � � �� � � Telegraph equation � � � � � � Resistance R Inductance L Capacitance C Conductance G Wiki/Pixabay Voltage V Current I ∂ V ∂ x ( x , t ) = − L · ∂ I ∂ t ( x , t ) − R · I ( x , t ) ∂ x ( x , t ) = − C · ∂ V ∂ I ∂ t ( x , t ) − G · V ( x , t ) or V xx − LC · V tt − ( RC + GL ) · V t − GR · V = 0 � �� � � �� � Wave equation Effect of losses

  3. Six–vertex model O H O H O H O H O Square grid with O in the vertices H H H H H and H on the edges. O H O H O H O H O H H H H H O H O H O H O H O H H H H H O H O H O H O H O H H H H H O H O H O H O H O

  4. Six–vertex model Square grid with O in the vertices O H O H O H O H O and H on the edges. H H H H H O H O H O H O H O Finite/infinite domain. H H H H H O H O H O H O H O H H H H H O H O H O H O H O H H H H H O H O H O H O H O

  5. Six–vertex model Square grid with O in the vertices O H O H O H O H O and H on the edges. H H H H H O H O H O H O H O Finite/infinite domain. H H H H H Configurations: possible matchings O H O H O H O H O of all atoms inside domain into H H H H H H 2 O molecules. O H O H O H O H O This is square ice model . H H H H H Real-world ice has somewhat similar O H O H O H O H O (although 3d) structure.

  6. Six–vertex model Square grid with O in the vertices O H O H O H O H O and H on the edges. H H H H H O H O H O H O H O Finite/infinite domain. H H H H H Configurations: possible matchings O H O H O H O H O of all atoms inside domain into H H H H H H 2 O molecules. O H O H O H O H O This is square ice model . H H H H H Real-world ice has somewhat similar O H O H O H O H O (although 3d) structure. Also known as six vertex model . H H H H O O H O H H O O H O H H H H

  7. Six–vertex model: Gibbs measures H H H H O O H O H H O O H O H H H H a 1 a 2 b 1 b 2 c 1 c 2 Statistical mechanics starting from (Lieb–67): O H O H O H O H O Assign Gibbs weights H H H H H O H O H O H O H O a #( a 1 ) a #( a 2 ) b #( b 1 ) b #( b 2 ) c #( c 1 ) c #( c 2 ) 1 2 1 2 1 2 H H H H H Z ( a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ) O H O H O H O H O H H H H H [ Depends only on b 1 b 2 a 1 a 2 and c 1 c 2 a 1 a 2 .] O H O H O H O H O H H H H H Asymptotic properties of O H O H O H O H O Gibbs measures?

  8. Six–vertex model: Gibbs measures H H H H O O H O H H O O H O H H H H a 1 a 2 b 1 b 2 c 1 c 2 O H O H O H O H O Assign Gibbs weights H H H H H O H O H O H O H O a #( a 1 ) a #( a 2 ) b #( b 1 ) b #( b 2 ) c #( c 1 ) c #( c 2 ) 1 2 1 2 1 2 H H H H H Z ( a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ) O H O H O H O H O H H H H H [ Depends only on b 1 b 2 a 1 a 2 and c 1 c 2 a 1 a 2 .] O H O H O H O H O H H H H H Asymptotic properties of O H O H O H O H O Gibbs measures? Understanding is still very limited.

  9. Question for today Telegraph equation Six–vertex model O H O H O H O H O H H H H H O H O H O H O H O H H H H H � � �� � � O H O H O H O H O � � � � � � V xx − V tt − α V t − β V x − γ V = 0 H H H H H O H O H O H O H O H H H H H O H O H O H O H O a #( a 1) a #( a 2) b #( b 1) b #( b 2) c #( c 1) c #( c 2) 1 2 1 2 1 2 Z ( a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ) What do they have in common?

  10. Stochastic six–vertex model a 1 a 2 b 1 b 2 c 1 c 2 H H H O H H O O H H O O H O H H H H An equivalent representation Collection of paths on the plane

  11. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 H H H O H H O O H H O O H O H H H H Assumption: a 1 = a 2 = 1 , b 1 + c 1 = 1 , b 2 + c 2 = 1 (Gwa–Spohn–92) � � ∆ = a 1 a 2 + b 1 b 2 − c 1 c 2 Implies ≥ 1 2 √ a 1 a 2 b 1 b 2

  12. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Take arbitrary boundary conditions in the quadrant . . . 5 4 3 2 1 0 4 5 1 2 3 . . .

  13. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . b 1 5 4 choice 3 2 1 − b 1 1 . . . 0 3 4 5 1 2

  14. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . 5 4 no choice 3 2 1 0 3 4 5 1 2 . . .

  15. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . 5 4 no choice 3 2 1 0 3 4 5 1 2 . . .

  16. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . 5 4 no choice 3 2 1 0 3 4 5 1 2 . . .

  17. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . b 2 5 4 choice 3 2 1 − b 2 1 . . . 0 3 4 5 1 2

  18. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . 5 4 no choice 3 2 1 0 3 4 5 1 2 . . .

  19. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . b 1 5 4 choice 3 2 1 − b 1 1 . . . 0 3 4 5 1 2

  20. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . b 2 5 4 choice 3 2 1 − b 2 1 . . . 0 3 4 5 1 2

  21. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . b 2 5 4 choice 3 2 1 − b 2 1 . . . 0 3 4 5 1 2

  22. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Proceed with sequential stochastic sampling . . . b 1 5 4 choice 3 2 1 − b 1 1 . . . 0 3 4 5 1 2

  23. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 Until the quadrant is filled . . . 5 4 3 2 1 0 4 5 1 2 3 . . .

  24. Stochastic six–vertex model 1 1 b 1 b 2 1 − b 1 1 − b 2 The resulting paths are level lines of the height function . . . 3 2 5 2 1 4 • Height is 0 at the origin, 3 • increases up, 1 0 • decreases to the 2 right. − 1 1 0 − 1 − 2 0 4 5 1 2 3 . . .

  25. Domain–wall and fixed weights . . . b 1 b 2 5 4 3 2 2 5 s = 1 − b 1 1 − b 2 4 3 2 2 1 4 1 1 1 3 2 2 3 LH ( Lx , Ly ) → h ( x , y ) =  2 1 1 0 0 x y > s − 1 , 0 , 2   ( √ s x − √ y ) 2   y ≤ s − 1 s ≤ x , 1 1 1 0 0 1 − s 1    x  y − x , y < s . 4 0 3 5 . . . 1 2 Theorem. (Borodin–Corwin–Gorin-14) For domain–wall boundary conditions and fixed 0 < b 2 < b 1 < 1 , 1 L H ( Lx , Ly ) → h ( x , y ) with fluctuations on L 1 / 3 scale given by the Tracy–Widom distribution. TW = universal law for the largest eigenvalue of Hermitian matrices and for particle system in Kardar–Parisi–Zhang class

  26. Domain–wall and fixed weights . . . b 1 b 2 5 4 3 2 2 5 s = 1 − b 1 1 − b 2 4 3 2 2 1 4 1 1 1 3 2 2 3 LH ( Lx , Ly ) → h ( x , y ) =  2 1 1 0 0 x y > s − 1 , 0 , 2   ( √ s x − √ y ) 2   y ≤ s − 1 s ≤ x , 1 1 1 0 0 1 − s 1    x  y − x , y < s . 4 0 3 5 . . . 1 2 Theorem. (Borodin–Corwin–Gorin-14) For domain–wall boundary conditions and fixed 0 < b 2 < b 1 < 1 , 1 L H ( Lx , Ly ) → h ( x , y ) with fluctuations on L 1 / 3 scale given by the Tracy–Widom distribution. ρ x · s + ρ y · ( s + ( s − 1) ρ ) 2 = 0 , ρ = h x . 1st–order non-linear PDE (Gwa–Spohn–92) (Reshetikhin–Sridhar–16)

  27. Domain–wall and fixed weights . . . b 1 b 2 5 4 3 2 2 5 s = 1 − b 1 1 − b 2 4 3 2 2 1 4 1 1 1 3 2 2 3 LH ( Lx , Ly ) → h ( x , y ) =  2 1 1 0 0 x y > s − 1 , 0 , 2   ( √ s x − √ y ) 2   y ≤ s − 1 s ≤ x , 1 1 1 0 0 1 − s 1    x  y − x , y < s . 4 0 3 5 . . . 1 2 ρ x · s + ρ y · ( s + ( s − 1) ρ ) 2 = 0 , ρ = h x • Instead of b 1 , b 2 , only s . Where is the second parameter? • Where is the linear telegraph equation?

  28. Domain–wall and fixed weights . . . b 1 b 2 5 4 3 2 2 5 s = 1 − b 1 1 − b 2 4 3 2 2 1 4 1 1 1 3 2 2 3 LH ( Lx , Ly ) → h ( x , y ) =  2 1 1 0 0 x y > s − 1 , 0 , 2   ( √ s x − √ y ) 2   y ≤ s − 1 s ≤ x , 1 1 1 0 0 1 − s 1    x  y − x , y < s . 4 0 3 5 . . . 1 2 ρ x · s + ρ y · ( s + ( s − 1) ρ ) 2 = 0 , ρ = h x • Instead of b 1 , b 2 , only s . Where is the second parameter? • Where is the linear telegraph equation? (Borodin–Gorin–18): One needs to rescale weights b 1 , b 2 .

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