Stochastik AG

Rongfeng Sun (Singapore) und Jan Swart (Prag)

Beide Vorträge finden im Rahmen des Workshops vom 12.12.-13.12.2019 im H13 statt.

Rongfeng Sun
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The 2-dimensional KPZ equation in the entire subcritical regime}\end{center}

We consider the KPZ equation in space dimension 2 driven by space-time white noise. Previously, we showed that if the noise is mollified in space on scale epsilon and the noise strength scaled appropriately, then a phase transition occurs. Chatterjee and Dunlap then showed that the solution admits non-trivial sub-sequential scaling limits deep within the subcritical regime. In recent joint work with F. Caravenna and N. Zygouras, we showed that the limit exists in the entire subcritical regime, and we identify it as the solution of an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations. Deep within the subcritical regime, the same result was obtained independently by Y. Gu.

Jan Swart
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Frozen percolation on the 3-regular tree}\end{center}

In frozen percolation, i.i.d. uniformly distributed activation times are
assigned to the edges of a graph. At its assigned time, an edge opens provided
neither of its endvertices is part of an infinite open cluster; in the
opposite case, it freezes. David Aldous (2000) showed that such a process can
be constructed on the infinite 3-regular tree and asked whether the event that
a given edge freezes is a measurable function of the activation times assigned
to all edges.

In joint work with Antar Bandyopadhyay (2005), Aldous showed that the question
can be reformulated in terms of a „Recursive Tree Process“ (RTP), which is a
sort of Markov chain in which time has a tree-like structure and the state of
each vertex is a function of its descendants and some i.i.d. randomness. They
called such an RTP endgenous if the state at the root is is a measurable
function of the i.i.d. randomness attached to the vertices and showed that
endogeny is equivalent to the statement that a certain bivariate recursive
distributional equation has a unique solution.

In joint work with Balázs Ráth and Tamás Terpai, we were able to settle
Aldous‘ original question by showing that for frozen percolation on the binary
tree, bivariate uniqueness does not hold. An essential role in our proofs is
played by a frozen percolation process on a continuous-time binary Galton
Watson tree that has nice scale invariant properties.