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Non Unitary Random Walks. Philippe Jacquet INRIA-Polytechnique. In 1976…. Cadillac produced its last dinosaur…. Philippe was in INRIA for his first job creating the Algorithm project. Black Hole information loss paradox. Hawking’s information loss paradox claim (1976)

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Non unitary random walks l.jpg

Non Unitary Random Walks

Philippe Jacquet


In 1976 l.jpg
In 1976…

Cadillac produced its last dinosaur…

Black hole information loss paradox l.jpg
Black Hole information loss paradox Algorithm project

Hawking’s information loss paradox claim (1976)

Black hole are not unitary and can destroye

Information (hyper-computing, « P=NP », retro-information)

Unitary universe l.jpg
Unitary universe Algorithm project

In an (honest) unitary universe

All probabilities sums to one (!).

Take a rabbit

One month later it is either:

0.5 or 0.5


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Unitary universe Algorithm project

Wave function Ψis quantum physics

Probabilistic interpretation

Unitarity is a physical assumption

Non unitary universe l.jpg
Non-unitary universe Algorithm project

Take a and a black hole

One month later either

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In 2006 Algorithm project

General motors produced its last dinosaur:

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2006 Information loss paradox Algorithm project

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

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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

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Model refinement economics

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


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non unitary Markovian system economics

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


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Toward a refutation of Hawking refutation? economics

  • 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


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Non unitarity Markov probability distorsion economics

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 :

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Apparent Black Hole repulsion economics

The rabbit is repelled from Black Hole with apparent probability :

if , then the repulsion is 100%

if , then the repulsion is just increased

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Non unitary random walk economics

unbounded random walk

sums of probabilities of state n for BH lifetime t


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

  • With

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Functional equation resolution walks:

  • We have

    • But

    • With



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Analytic resolution with Kernel walks:

  • R(z,u) is analytic in the unit disk beyond

    • thus

      • And

      • Singular for and

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Asymptotic behavior walks:

  • Let

    If then main singularity on

    • relative repulsion

      • Uniform on all ranges

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Large unitary default rate walks:

  • When then main singularity at

    • The relative repulsion is neutral (flat)

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Nice plots walks:

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Generalization walks:

  • Non uniform random walks

    • Gravitational walks


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Non unitary Gravitational walks walks:

Repulsion point

Apparent potential for lifetime=1, 400,800, 1600

Actual potential

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2D non unitary gravitational walks walks:

Repulsion ring

Apparent potential

Actual potential