99 research outputs found
The FKG Inequality and Some Monotonicity Properties of Partial Orders
Let (a1 , . . . , am, b1, . . . , bn) be a random permutation of 1, 2, . . ., m + n. Let P be a partial order on the a’s and b’s involving only inequalities of the form ai \u3c aj or bi \u3c bj, and let P\u27 be an extension of P to include inequalities of the form ai \u3c bj; i.e, P\u27 = P ∪ P\u27\u27, where P\u27\u27 involves only inequalities of the form ai \u3c bj. We prove the natural conjecture of R. L. Graham, A. C. Yao, and F. F. Yao [SIAM J. Alg. Discr. Meth. 1 (1980), pp. 251–258] that in particular (*) Pr (a1 \u3c b1|P\u27) ≥ Pr (a1 \u3c b1|P). We give a simple example to show that the more general inequality (*) where P is allowed to contain inequalities of the form ai \u3c bj is false. This is surprising because as Graham, Yao, and Yao proved, the general inequality (*) does hold if P totally orders both the a’s and the b’s separately. We give a new proof of the latter result. Our proofs are based on the FKG inequality
Some correlation inequalities for probabilistic analysis of algorithms
The analysis of many randomized algorithms, for example in dynamic load balancing, probabilistic divide-and-conquer paradigm and distributed edge-coloring, requires ascertaining the precise nature of the correlation between the random variables arising in the following prototypical ``balls-and-bins'' experiment. Suppose a certain number of balls are thrown uniformly and independently at random into bins. Let be the random variable denoting the number of balls in the th bin, . These variables are clearly not independent and are intuitively negatively related. We make this mathematically precise by proving the following type of correlation inequalities: \begin{itemize} \item For index sets such that or , and any non--negative integers , \prob[\sum_{i \in I} X_i \geq t_I \mid \sum_{j \in J} X_j \geq t_J] \-5mm] \[\leq \prob[\sum_{i \in I} X_i \geq t_I] . \item For any disjoint index sets , any and any non--negative integers and , \prob[\bigwedge_{i \in I}X_i \geq t_i \mid \bigwedge_{j \in J} X_j \geq t_j]\-5mm]\[ \leq \prob[\bigwedge_{i \in I'}X_i \geq t_i \mid \bigwedge_{j \in J'} X_j \geq t_j] . \end{itemize} Although these inequalities are intuitively appealing, establishing them is non--trivial; in particular, direct counting arguments become intractable very fast. We prove the inequalities of the first type by an application of the celebrated FKG Correlation Inequality. The proof for the second uses only elementary methods and hinges on some {\em monotonicity} properties. More importantly, we then introduce a general methodology that may be applicable whenever the random variables involved are negatively related. Precisely, we invoke a general notion of {\em negative assocation\/} of random variables and show that: \begin{itemize} \item The variables are negatively associated. This yields most of the previous results in a uniform way. \item For a set of negatively associated variables, one can apply the Chernoff-Hoeffding bounds to the sum of these variables. This provides a tool that facilitates analysis of many randomized algorithms, for example, the ones mentioned above
Mixing time of the adjacent walk on the simplex
By viewing the -simplex as the set of positions of ordered particles
on the unit interval, the adjacent walk is the continuous time Markov chain
obtained by updating independently at rate 1 the position of each particle with
a sample from the uniform distribution over the interval given by the two
particles adjacent to it. We determine its spectral gap and prove that both the
total variation distance and the separation distance to the uniform
distribution exhibit a cutoff phenomenon, with mixing times that differ by a
factor . The results are extended to the family of log-concave distributions
obtained by replacing the uniform sampling by a symmetric log-concave Beta
distribution
Existence of a tricritical point for the Blume-Capel model on
We prove the existence of a tricritical point for the Blume-Capel model on
for every . The proof in relies on a novel
combinatorial mapping to an Ising model on a larger graph, the techniques of
Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the
celebrated infrared bound. In , the proof relies on a quantitative
analysis of crossing probabilities of the dilute random cluster representation
of the Blume-Capel. In particular, we develop a quadrichotomy result in the
spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to
obtain a fine picture of the phase diagram in , including asymptotic
behaviour of correlations in all regions. Finally, we show that the techniques
used to establish subcritical sharpness for the dilute random cluster model
extend to any .Comment: 55 pages. 4 figures. v2 includes fixes of typos and clarification
Random structures for partially ordered sets
This thesis is presented in two parts. In the first part, we study a family of models
of random partial orders, called classical sequential growth models, introduced by
Rideout and Sorkin as possible models of discrete space-time. We analyse a particular
model, called a random binary growth model, and show that the random partial
order produced by this model almost surely has infinite dimension. We also give
estimates on the size of the largest vertex incomparable to a particular element of
the partial order. We show that there is some positive probability that the random
partial order does not contain a particular subposet. This contrasts with other existing
models of partial orders. We also study "continuum limits" of sequences of
classical sequential growth models. We prove results on the structure of these limits
when they exist, highlighting a deficiency of these models as models of space-time.
In the second part of the thesis, we prove some correlation inequalities for mappings
of rooted trees into complete trees. For T a rooted tree we can define the proportion
of the total number of embeddings of T into a complete binary tree that map the
root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and
Morayne states that, for two binary trees with one a subposet of the other, this
proportion is larger for the larger tree. They conjecture that the same is true for
two arbitrary trees with one a subposet of the other. We disprove this conjecture
by analysing the asymptotics of this proportion for large complete binary trees.
We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a
correlation inequality which enables us to generalise their result in other directions
A dichotomy theory for height functions
Height functions are random functions on a given graph, in our case
integer-valued functions on the two-dimensional square lattice. We consider
gradient potentials which (informally) lie between the discrete Gaussian and
solid-on-solid model (inclusive). The phase transition in this model, known as
the roughening transition, Berezinskii-Kosterlitz-Thouless transition, or
localisation-delocalisation transition, was established rigorously in the 1981
breakthrough work of Fr\"ohlich and Spencer. It was not until 2005 that
Sheffield derived continuity of the phase transition. First, we establish
sharpness, in the sense that covariances decay exponentially in the localised
phase. Second, we show that the model is delocalised at criticality, in the
sense that the set of potentials inducing localisation is open in a natural
topology. Third, we prove that the pointwise variance of the height function is
at least in the delocalised regime, where is the distance to the
boundary, and where denotes a universal constant. This implies that the
effective temperature of any potential cannot lie in the interval
(whenever it is well-defined), and jumps from to at least at the
critical point. We call this range of forbidden values the effective
temperature gap.Comment: 68 pages, 20 figures; added definition of correlation length and
improved presentatio
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