Euler Walks

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Euler Walks

• Anatoly Patrick
• Laboratory of Theoretical Physics
• Joint Institute for Nuclear Research
• Dubna

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Passive Walks

• Simple 2D Random Walk

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Active Walks

• Euler Walk

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• Typical features
• The domain of visited edges is not compact.
• After T steps its size

Simple 2D Random Walk
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• Typical features
• The domain of visited edges is compact.
• After T steps its size

Euler Walk
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• Euler walk on a Cayley tree.

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• Let Tn, n 1,2,3,... be the time of the n-th
  return to the origin.
• Then

is a positive and uniformly integrable
martingale.
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• What is a martingale?
• Consider a sequence of random variables X1, X2,
  X3,.
• Let, for all n, E Xn lt ?, and
• E Xn1 Xn, Xn-1,, X1 Xn.
• Then the sequence X1, X2, X3, is called a
  martingale.

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• What is a conditional expected value
• E X Y ?
• E X Y is a random variable.
• In fact, it is some function f (Y ).
• Namely, f (z) E X Y z.
• For instance, if X can be expressed as
• X G (U,Y ),
• where U is another random variable independent
  of Y,
• then f (z) E G (U,z ) .

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• Corollary 1.

where Y is a random variable with a proper
distribution, EY 2.
Corollary 2.
The End.