Thus, conditioned on ξ , the law of S is given by a Bernoulli percolation on Λ, with an edge belonging to S with probability 1 if e is unsatisﬁed and with probability R ∞ 2|Ke | e−xdx = e−2|Ke | if e is satisﬁed.
Random Surfaces
On the other hand, given G[W ], t and t′ are independent r.v.’s, each with law Bin(W, p) (where we say t = (tw : w ∈ W ) has “law Bin(W, p)” if the tw ’s are independent, mean p Bernoullis).
Factors in random graphs
We say that S is a percolation cluster when P is the law of the inﬁnite cluster in super–critical Bernoulli site– percolation on Zd .
Recurrence and transience for long-range reversible random walks on a random point process
It is known (see and ) that for each bounded Borel set B ⊂ Rd the number of points S (B ) has the same law of the sum Pi Bi , Bi ’s being independent Bernoulli random variables with parameters λi (B ).
Recurrence and transience for long-range reversible random walks on a random point process
Then, using matricial freeness, we obtain again the s-free convolution of two collective Bernoulli laws, κp,q and κq ,p .
Limit distributions of random matrices
In contrast to the ﬁrst Kelvin/Helmhotz theorem associated with Eq. (26) we have no simple advantage from this second vorticity law; in particular, the equations of motion for r and vλ have to be solved, in addition to the conservation laws for s and qλ and (not shown here) the Bernoulli equation for S .
Potential equations for plasmas round a rotating black hole
***