160 research outputs found

### Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new percolation model to describe and analyze the spread of an
epidemic on a general directed and locally finite graph. We assign a
two-dimensional random weight vector to each vertex of the graph in such a way
that the weights of different vertices are i.i.d., but the two entries of the
vector assigned to a vertex need not be independent. The probability for an
edge to be open depends on the weights of its end vertices, but conditionally
on the weights, the states of the edges are independent of each other. In an
epidemiological setting, the vertices of a graph represent the individuals in a
(social) network and the edges represent the connections in the network. The
weights assigned to an individual denote its (random) infectivity and
susceptibility, respectively. We show that one can bound the percolation
probability and the expected size of the cluster of vertices that can be
reached by an open path starting at a given vertex from above and below by the
corresponding quantities for respectively independent bond and site percolation
with certain densities; this generalizes a result of Kuulasmaa. Many models in
the literature are special cases of our general model.Comment: 15 page

### Assessing forensic evidence by computing belief functions

We first discuss certain problems with the classical probabilistic approach
for assessing forensic evidence, in particular its inability to distinguish
between lack of belief and disbelief, and its inability to model complete
ignorance within a given population. We then discuss Shafer belief functions, a
generalization of probability distributions, which can deal with both these
objections. We use a calculus of belief functions which does not use the much
criticized Dempster rule of combination, but only the very natural
Dempster-Shafer conditioning. We then apply this calculus to some classical
forensic problems like the various island problems and the problem of parental
identification. If we impose no prior knowledge apart from assuming that the
culprit or parent belongs to a given population (something which is possible in
our setting), then our answers differ from the classical ones when uniform or
other priors are imposed. We can actually retrieve the classical answers by
imposing the relevant priors, so our setup can and should be interpreted as a
generalization of the classical methodology, allowing more flexibility. We show
how our calculus can be used to develop an analogue of Bayes' rule, with belief
functions instead of classical probabilities. We also discuss consequences of
our theory for legal practice.Comment: arXiv admin note: text overlap with arXiv:1512.01249. Accepted for
publication in Law, Probability and Ris

### Uniquely determined uniform probability on the natural numbers

In this paper, we address the problem of constructing a uniform probability
measure on $\mathbb{N}$. Of course, this is not possible within the bounds of
the Kolmogorov axioms and we have to violate at least one axiom. We define a
probability measure as a finitely additive measure assigning probability $1$ to
the whole space, on a domain which is closed under complements and finite
disjoint unions. We introduce and motivate a notion of uniformity which we call
weak thinnability, which is strictly stronger than extension of natural
density. We construct a weakly thinnable probability measure and we show that
on its domain, which contains sets without natural density, probability is
uniquely determined by weak thinnability. In this sense, we can assign uniform
probabilities in a canonical way. We generalize this result to uniform
probability measures on other metric spaces, including $\mathbb{R}^n$.Comment: We added a discussion of coherent probability measures and some
explanation regarding the operator we study. We changed the title to a more
descriptive one. Further, we tidied up the proofs and corrected and
simplified some minor issue

### Phase transition and uniqueness of levelset percolation

The main purpose of this paper is to introduce and establish basic results of
a natural extension of the classical Boolean percolation model (also known as
the Gilbert disc model). We replace the balls of that model by a positive
non-increasing attenuation function $l:(0,\infty) \to (0,\infty)$ to create the
random field $\Psi(y)=\sum_{x\in \eta}l(|x-y|),$ where $\eta$ is a homogeneous
Poisson process in ${\mathbb R}^d.$ The field $\Psi$ is then a random potential
field with infinite range dependencies whenever the support of the function $l$
is unbounded.
In particular, we study the level sets $\Psi_{\geq h}(y)$ containing the
points $y\in {\mathbb R}^d$ such that $\Psi(y)\geq h.$ In the case where $l$
has unbounded support, we give, for any $d\geq 2,$ exact conditions on $l$ for
$\Psi_{\geq h}(y)$ to have a percolative phase transition as a function of $h.$
We also prove that when $l$ is continuous then so is $\Psi$ almost surely.
Moreover, in this case and for $d=2,$ we prove uniqueness of the infinite
component of $\Psi_{\geq h}$ when such exists, and we also show that the
so-called percolation function is continuous below the critical value $h_c$.Comment: 25 page

### Bounds for avalanche critical values of the Bak-Sneppen model

We study the Bak-Sneppen model on locally finite transitive graphs $G$, in
particular on Z^d and on T_Delta, the regular tree with common degree Delta. We
show that the avalanches of the Bak-Sneppen model dominate independent site
percolation, in a sense to be made precise. Since avalanches of the Bak-Sneppen
model are dominated by a simple branching process, this yields upper and lower
bounds for the so-called avalanche critical value $p_c^{BS}(G)$. Our main
results imply that 1/(Delta+1) <= \leq p_c^{BS}(T_Delta) \leq 1/(Delta -1)$,
and that$1/(2d+1)\leq p_c^{BS}(Z^d)\leq 1/(2d)+ 1/(2d)^2+O(d^{-3}), as
d\to\infty.Comment: 19 page

### Long-range percolation on the hierarchical lattice

We study long-range percolation on the hierarchical lattice of order $N$,
where any edge of length $k$ is present with probability
$p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed
$\beta$, we show that the critical value $\alpha_c(\beta)$ is non-trivial if
and only if $N < \beta < N^2$. Furthermore, we show uniqueness of the infinite
component and continuity of the percolation probability and of
$\alpha_c(\beta)$ as a function of $\beta$. This means that the phase diagram
of this model is well understood.Comment: 24 page

### Existence and uniqueness of the stationary measure in the continuous Abelian sandpile

Let \Lambda be a finite subset of Z^d. We study the following sandpile model
on \Lambda. The height at any given vertex x of \Lambda is a positive real
number, and additions are uniformly distributed on some interval [a,b], which
is a subset of [0,1]. The threshold value is 1; when the height at a given
vertex exceeds 1, it topples, that is, its height is reduced by 1, and the
heights of all its neighbours in \Lambda increase by 1/2d. We first establish
that the uniform measure \mu on the so called "allowed configurations" is
invariant under the dynamics. When a < b, we show with coupling ideas that
starting from any initial configuration of heights, the process converges in
distribution to \mu, which therefore is the unique invariant measure for the
process. When a = b, that is, when the addition amount is non-random, and a is
rational, it is still the case that \mu is the unique invariant probability
measure, but in this case we use random ergodic theory to prove this; this
proof proceeds in a very different way. Indeed, the coupling approach cannot
work in this case since we also show the somewhat surprising fact that when a =
b is rational, the process does not converge in distribution at all starting
from any initial configuration.Comment: 22 page

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