Probabilistic Proof Research Articles

A 2-spine decomposition of the critical Galton-Watson tree and a probabilistic proof of Yaglom’s theorem

In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yaglom’s theorem.

Random surface growth and Karlin-McGregor polynomials

We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered in the seminal work of Borodin and Olshanski in [10] and the ones on the BC-type graph recently studied by Cuenca in [17]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan [8] and Cerenzia [15], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan [16]. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski in [11], and that depend on the inhomogeneities.

Intertwining of the Wright-Fisher diffusion

It is known that the time until a birth and death process reaches a certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. Their coupling is of a special type called intertwining of Markov processes. We apply this technique to couple the Wright-Fisher diffusion with reflection at 1/2 and a pure birth process. We show that in our coupling the time of absorption of the diffusion is a.s. equal to the time of explosion of the pure birth process. The coupling also allows us to interpret the diffusion as being initially reluctant to get absorbed, but later getting more and more compelled to get absorbed.

Counting Walks and Graph Homomorphisms via Markov Chains and Importance Sampling

Hoffman [7] proved a matrix inequality that yields a useful upper bound on the number of walks in a graph. Sidorenko [14] extended the bound on the number of walks to a bound on the number of homomorphisms from a tree to a graph. In this expository note, we give a short probabilistic proof of both results, using the basic identity of importance sampling.

Some Probabilistic Interpretations of the Multinomial Theorem

SummaryWe give a simple probabilistic proof of an important combinatorial identity. In the process, we show via probabilistic arguments that there are exactly terms in the multinomial expansion of (λ1 + λ2 + ⋯ + λm)n. Also, an alternate probabilistic proof of the multinomial theorem is obtained using the convolution property of the Poisson distribution.

A Mean Field Limit for the Vlasov–Poisson System

We present a probabilistic proof of the mean field limit and propagation of chaos N-particle systems in three dimensions with positive (Coulomb) or negative (Newton) 1/r potentials scaling like 1/N and an N-dependent cut-off which scales like $${N^{-1/3+ \epsilon}}$$ . In particular, for typical initial data, we show convergence of the empirical distributions to solutions of the Vlasov–Poisson system with either repulsive electrical or attractive gravitational interactions.

Optimal distributed interconnectivity of multi-robot systems by spatially-constrained clustering

A spatially-constrained clustering algorithm is presented in this paper. This algorithm is a distributed clustering approach to fine-tune the optimal distances between agents of the system to strengthen the data passing among them using a set of spatial constraints. In fact, this method will increase interconnectivity among agents and clusters, leading to improvement of the overall communicative functionality of the multi-robot system. This strategy will lead to the establishment of loosely-coupled connections among the clusters. These implicit interconnections will mobilize the clusters to receive and transmit information within the multi-agent system. In other words, this algorithm classifies each agent into the clusters with the lowest cost of local communication with its peers. This research demonstrates that the presented decentralized method will actually boost the communicative agility of the swarm by probabilistic proof of the acquired optimality. Hence, the common assumption regarding the full-knowledge of the agents’ primary locations has been fully relaxed compared to former methods. Consequently, the algorithm’s reliability and efficiency is confirmed. Furthermore, the method’s efficacy in passing information will improve the functionality of higher-level swarm operations, such as task assignment and swarm flocking. Analytical investigations and simulated accomplishments, corresponding to highly-populated swarms, prove the claimed efficiency and coherence.

The Local Cut Lemma

The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos (2010) have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm, Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealing to the LLL. The aim of this paper is to provide a generalization of the LLL which implies these new combinatorial results. This generalization, which we call the Local Cut Lemma, concerns a random cut in a directed graph with certain properties. Note that our result has a short probabilistic proof that does not use entropy compression. As a consequence, it not only shows that a certain probability is positive, but also gives an explicit lower bound for this probability. As an illustration, we present a new application (an improved lower bound on the number of edges in color-critical hypergraphs) as well as explain how to use the Local Cut Lemma to derive some of the results obtained previously using the entropy compression method.

Solving dense symmetric indefinite systems using GPUs

SummaryThis paper studies the performance of different algorithms for solving a dense symmetric indefinite linear system of equations on multicore CPUs with a Graphics Processing Unit (GPU). To ensure the numerical stability of the factorization, pivoting is required. Obtaining high performance of such algorithms on the GPU is difficult because all the existing pivoting strategies lead to frequent synchronizations and irregular data accesses. Until recently, there has not been any implementation of these algorithms on a hybrid CPU/GPU architecture. To improve their performance on the hybrid architecture, we explore different techniques to reduce the expensive data transfer and synchronization between the CPU and GPU, or on the GPU (e.g., factorizing the matrix entirely on the GPU or in a communication‐avoiding fashion). We also study the performance of the solver using iterative refinements along with the factorization without pivoting combined with the preprocessing technique based on random butterfly transformations, or with the mixed‐precision algorithm where the matrix is factorized in single precision. This randomization algorithm only has a probabilistic proof on the numerical stability, and for this paper, we only focused on the mixed‐precision algorithm without pivoting. However, they demonstrate that we can obtain good performance on the GPU by avoiding the pivoting and using the lower precision arithmetics, respectively. As illustrated with the application in acoustics studied in this paper, in many practical cases, the matrices can be factorized without pivoting. Because the componentwise backward error computed in the iterative refinement signals when the algorithm failed to obtain the desired accuracy, the user can use these potentially unstable but efficient algorithms in most of the cases and fall back to a more stable algorithm with pivoting only in the case of the failure. Copyright © 2017 John Wiley & Sons, Ltd.

The stochastic solution to a Cauchy problem for degenerate parabolic equations

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ∈(0,1], the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement δ≥1/2. Uniqueness results, including a Feynman–Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.

The Energy of the Alphabet Model

We call \emph{Alphabet model} a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one dimensional lattice with an exchange dynamics. The rates of exchange are local but under suitable conditions the dynamics is reversible with a Gibbsian like invariant measure with long range interactions. We discuss geometrically the conditions of reversibility on a ring that correspond to a gradient condition on the graph of configurations or equivalently to a divergence free condition on a graph structure associated to the types of particles. We show that much of the information on the interactions between particles can be encoded in associated \emph{Tournaments} that are a special class of oriented directed graphs. In particular we show that the interactions of reversible models are corresponding to strongly connected tournaments. The possible minimizers of the energies are in correspondence with the Hamiltonian cycles of the tournaments. We can then determine how many and which are the possible minimizers of the energy looking at the structure of the associated tournament. As a byproduct we obtain a probabilistic proof of a classic Theorem of Camion \cite{Camion} on the existence of Hamiltonian cycles for strongly connected tournaments. Using these results we obtain in the case of an equal number of k types of particles new representations of the Hamiltonians in terms of translation invariant $k$-body long range interactions. We show that when $k=3,4$ the minimizer of the energy is always unique up to translations. Starting from the case $k=5$ it is possible to have more than one minimizer. In particular it is possible to have minimizers for which particles of the same type are not joined together in single clusters.

A probabilistic proof of some integral formulas involving the Meijer G-function

New integral formulas involving the Meijer G-function are derived using recent results concerning distributional characterisations and distributional transformations in probability theory.

Branching Brownian motion, mean curvature flow and the motion of hybrid zones

We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial Λ-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

Proofs of binomial and geometric series expansions under a probabilistic microscope with real-life examples

This paper begins with a new and complete probabilistic proof of the well-known binomial theorem. Then, it broadens the same spirit for handling a useful infinite series of which the well-known geometric series forms a special case. Also included are applications of the theorems and real-life examples.

Extensive amenability and an application to interval exchanges

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group$\text{IET}$of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on$\text{IET}$and show that there are subgroups$G<\text{IET}$admitting no finitely supported measure with trivial boundary.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes.

Probabilistic Proofs of a Binomial Identity, Its Inverse, and Generalizations

We give elementary probabilistic proofs of a binomial identity, its binomial inverse, and generalizations of both of these. The proofs are obtained by interpreting the sides of each identity as the probability of an event in two different ways. Each proof uses a classic balls-and-jars scenario.

Erratum to: A Probabilistic Proof of the Fundamental Gap Conjecture Via the Coupling by Reflection

Erratum to: A Probabilistic Proof of the Fundamental Gap Conjecture Via the Coupling by Reflection

A Probabilistic Proof of the Multinomial Theorem

In this note we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic probability concepts.

Valuing American options by simulation: A BSDEs approach

We provide probabilistic proofs of convergence of several easy to implement schemes for computing the value function of American (call and put) options written on a dividend paying stock governed by the geometric Brownian motion. The proofs are based on representations of the value function by means of solutions of some backward stochastic differential equations. Despite the probabilistic nature of the proofs the numerical schemes are nevertheless deterministic. Simulation results are also presented.