Non Unitary Random Walks

1. Non Unitary Random Walks Philippe Jacquet INRIA-Polytechnique

2. In 1976… Cadillac produced its last dinosaur…

3. Philippe was in INRIA for his first job creating the Algorithm project

4. Black Hole information loss paradox Hawking’s information loss paradox claim (1976) Black hole are not unitary and can destroye Information (hyper-computing, « P=NP », retro-information)

5. Unitary universe In an (honest) unitary universe All probabilities sums to one (!). Take a rabbit One month later it is either: 0.5 or 0.5 0.5+0.5=1

6. Unitary universe Wave function Ψis quantum physics Probabilistic interpretation Unitarity is a physical assumption

7. Non-unitary universe Take a and a black hole One month later either

8. In 2006 General motors produced its last dinosaur:

9. Philippe in his second job…

10. 2006 Information loss paradox Hawking refutes his 1996 argument: Probability inside blackhole decays with blackhole lifetime, probability away from blackhole remains constant At the end only probability outside black hole remain: The rabbit never falls in black hole The black hole never forms

11. Information loss paradox refutation explained in modern economics The Rabbit has $1000 He invests $500 in a modern bank And keeps $500 in his pocket The modern bank loses 30% per year: At the end the rabbit has most of its remaining money in his pocket

12. Model refinement Investment portfolio: The rabbit put every year q fraction of its pocket money in the bank at time t, Let the money in bank BH Let the money outside the bank BH BH

13. non unitary Markovian system Matrix R is not unitary Probability vector if , then at time t most money is in the pocket if , then at time t most money is in the bank Consequence: The money can still be move inside the black hole bank Hawking refutation is refutable BH

14. Toward a refutation of Hawking refutation? • At every moment (discret) • the rabbit has a non zero probability to fall in the Black Hole (gravity) • if , then the rabit may never fall in the BH • if , then the rabbit eventually falls in the BH BH

15. Non unitarity Markov probability distorsion Black Hole with remaining lifetime t a at time 0 Let the sum of probabilities for the rabbit inside BH Let the sum of probabilities for the rabbit outside BH Apparent attraction probability : Apparent repulsion probability :

16. Apparent Black Hole repulsion The rabbit is repelled from Black Hole with apparent probability : if , then the repulsion is 100% if , then the repulsion is just increased

17. Non unitary random walk unbounded random walk sums of probabilities of state n for BH lifetime t … BH

18. « Flajoleries » (Nice maths) about non unitary random walks:Bivariate functional equation • With

19. Functional equation resolution • We have • But • With 0 1

20. Analytic resolution with Kernel • R(z,u) is analytic in the unit disk beyond • thus • And • Singular for and

21. Asymptotic behavior • Let If then main singularity on • relative repulsion • Uniform on all ranges

22. Large unitary default rate • When then main singularity at • The relative repulsion is neutral (flat)

23. Nice plots

24. Generalization • Non uniform random walks • Gravitational walks … BH

25. Non unitary Gravitational walks Repulsion point Apparent potential for lifetime=1, 400,800, 1600 Actual potential

26. 2D non unitary random walks BH

27. 2D non unitary gravitational walks Repulsion ring Apparent potential Actual potential

28. Happy birthday Philippe!