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VERSION:2.0
PRODID:www.imperial.ac.uk
BEGIN:VEVENT
UID:641d98186b507
DTSTART:20230207T030000Z
SEQUENCE:0
TRANSP:OPAQUE
DTEND:20230207T040000Z
URL:https://www.imperial.ac.uk/events/157339/johannes-baumler-tu-munich-tba
/
LOCATION:140\, Huxley Building\, South Kensington Campus\, Imperial College
London\, London\, SW7 2AZ\, United Kingdom
SUMMARY:Johannes Bäumler (TU Munich): Chemical distances for long-range pe
rcolation
CLASS:PUBLIC
DESCRIPTION:This seminar will be presented in hybrid mode. The speaker wi
ll deliver his talk in person.\nTitle: Chemical distances for long-range p
ercolation \nAbstract: Consider long-range percolation on $\\mathbb{Z}^d$
\, where there is an edge between two points $x$ and $y$ with probability
asymptotic to $\\beta \\|x-y\\|^{-s}$\, independent of all other edges\, f
or some positive parameters $s$ and $\\beta$. In this talk\, we will focus
on the metric properties of the long-range percolation graph. The chemica
l distance between two points $x$ and $y$ is the number of steps one needs
to make in order to go from $x$ to $y$. For different values of $s$\, the
re are different regimes of how the chemical distance scales with the Eucl
idean distance. The transitions between these regimes happen at $s=d$ and
$s=2d$. After an overview of previous work\, we will focus on the case $s=
2d$. We will show that for $s=2d$\, for each dimension $d$ and for each $\
\beta > 0$\, there exists a $\\theta=\\theta(d\,\\beta) \\in (0\,1)$ such
that the chemical distance between $x$ and $y$ is of order $\\|x-y\\|^{\\t
heta}$. We will also discuss how the exponent $\\theta$ depends on the pa
rameter $\\beta$.\nThe talk will be followed by refreshments in the Huxley
Common Room at 4pm.
X-ALT-DESC;FMTTYPE=text/html:This seminar will be presented in hyb
rid mode. The speaker will deliver his talk in person.

\n#### T
itle: **Chemical distances for long-range percolation **<
/h4>\n

Abstract: Consider long-range percolation on $\\mathbb{Z}^d
$\, where there is an edge between two points $x$ and $y$ with probability
asymptotic to $\\beta \\|x-y\\|^{-s}$\, independent of all other edges\,
for some positive parameters $s$ and $\\beta$. In this talk\, we will focu
s on the metric properties of the long-range percolation graph. The chemic
al distance between two points $x$ and $y$ is the number of steps one need
s to make in order to go from $x$ to $y$. For different values of $s$\, th
ere are different regimes of how the chemical distance scales with the Euc
lidean distance. The transitions between these regimes happen at $s=d$ and
$s=2d$. After an overview of previous work\, we will focus on the case $s
=2d$. We will show that for $s=2d$\, for each dimension $d$ and for each $
\\beta > 0$\, there exists a $\\theta=\\theta(d\,\\beta) \\in (0\,1)$ such
that the chemical distance between $x$ and $y$ is of order $\\|x-y\\|^{\\
theta}$. We will also discuss how the exponent $\\theta$ dep
ends on the parameter $\\beta$.

\nThe talk will be foll
owed by refreshments in the Huxley Common Room at 4pm.

DTSTAMP:20230324T123120Z
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