Papers by Alberto Gandolfi
The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dim... more The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity λ > 0, with central charge c = 2λ. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary “layering vertex operators” Oβ with dimensions (∆,∆), with ∆ = λ 10(1− cosβ), that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions (∆ + k/3,∆ + k′/3), for all non-negative integers k, k′ satisfying |k − k′| = 0 mod 3. In this paper we introduce the edge counting field E(z) that counts the number of loop boundaries that pass close to the point z. We rigorously prove that the n-point functions of E are well defined and behave as expected for a conformal primary field with dimensions (1/3, 1/3). We analytically compute the four-point function 〈Oβ(z1)O−β(z2)E(z3)E(z4)〉 and ...
Communications in Mathematical Physics, 2021
We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as... more We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483–507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos ...
Journal of Mathematical Economics
Probability Theory and Related Fields
We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent pe... more We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on 7/e or zdx N with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ;ge or on a slab ;ge x {0,..., K } and prove both that percolation occurs and that the infinite component is unique for V = ~2 x {0, 1} or larger.
Here we solve the problem by a new formalization of probability theory based on a minimal collect... more Here we solve the problem by a new formalization of probability theory based on a minimal collection of axioms with additional context dependent conditions, whose overall consistency is then semantically verified. In Elementary Probability, i.e. probabilities involving boolean combinations of finitely many events, our theory leads to algebraization and, using Tarski Seidenberg reduction, to a proof of decidability of all problems. Inconsistency in Elementary Probability, on the other hand, is equivalent to, suitably redefined, arbitrage or Dutch Book. In the continuous case this leads to nonstandard analysis.
Dedicated to the memory of Thyagaraju Chelluri, a wonderful human being who would have become a f... more Dedicated to the memory of Thyagaraju Chelluri, a wonderful human being who would have become a fine mathematician had his life not been cut tragically short.
Sankhya A, 2012
We consider the classic problem of estimating T , the total number of species in a population, fr... more We consider the classic problem of estimating T , the total number of species in a population, from repeated counts in a simple random sample and look first at the Chao-Lee estimator: we initially show that such estimator can be obtained by reconciling two estimators of the unobserved probability, and then develop a sequence of improvements culminating in a Dirichlet prior Bayesian reinterpretation of the estimation problem. By means of this, we obtain simultaneous estimates of T , the normalized interspecies variance γ 2 and the parameter λ of the prior. Several simulations show that our estimation method is more flexible than several known methods we used as comparison; the only limitation, apparently shared by all other methods, seems to be that it cannot deal with the rare cases in which γ 2 > 1.
Dynamic Models of Infectious Diseases, 2013
We review several connections between percolation and SEIR epidemic models. In the first part we ... more We review several connections between percolation and SEIR epidemic models. In the first part we analyze the role of percolation in representing or approximating an epidemic model; in the second part we discuss the role of percolation in modelling the random networks on which the spread of the infectious diseases takes place. At the end, we propose a class of models which are mathematically treatable and at the same time incorporate most of the desirable features of epidemic modelling.
Key Engineering Materials, 2008
We present some examples of mathematical discoveries whose original import was mainly theoretical... more We present some examples of mathematical discoveries whose original import was mainly theoretical but which later ended up triggering extraordinary ad- vances in engineering, sometimes all the way down to technological realizations and market products. The examples we cite include Markov chains and Markov random fields, spin glasses, large deviations and the inverse conductivity problem, and their effects in various areas such as communication and imaging technologies.
Statistica Neerlandica, 2013
We develop a mathematical theory needed for moment estimation of the parameters in a general shif... more We develop a mathematical theory needed for moment estimation of the parameters in a general shifting level process (SLP) treating, in particular, the finite state space case geometric finite normal (GFN) SLP. For the SLP, we give expressions for the moment estimators together with asymptotic (co)variances, following, completing, and correcting CLINE (Journal of Applied Probability 20, 1983, 322-337); formulae are then made more explicit for the GFN-SLP. To illustrate the potential uses, we then apply the moment estimation method to a GFN-SLP model of array comparative genomic hybridization data. We obtain encouraging results in the sense that a segmentation based on the estimated parameters turns out to be faster than with other currently available methods, while being comparable in terms of sensitivity and specificity.
Probability Theory and Related Fields, 1989
The maximal value of the two-correlation for two-valued stationary one-dependent processes with f... more The maximal value of the two-correlation for two-valued stationary one-dependent processes with fixed probability e of a single symbol is determined. We show that the process attaining this bound is unique except when c~= 1/2, when there are exactly two different processes. The analogous problem for minimal two-correlation is discussed, and partial results are obtained.
Probability Theory and Related Fields, 1999
This paper studies a particular line in the parameter space of the FK random interaction random c... more This paper studies a particular line in the parameter space of the FK random interaction random cluster model for spin glasses following Katsura ([K]) and Mazza ([M]). We show that, after averaging over the random couplings, the occupied FK bonds have exactly a Bernoulli distribution. Comparison with explicit calculations on trees confirms the marginal role of FK percolation in determining phase transitions.
Milan Journal of Mathematics, 2005
Journal of Statistical Physics, 1991
Journal of Applied Probability, 1992
Consider a two-level storage system operating with the least recently used (LRU) or the first-in,... more Consider a two-level storage system operating with the least recently used (LRU) or the first-in, first-out (FIFO) replacement strategy. Accesses to the main storage are described by the independent reference model (IRM). Using the FKG inequality, we prove that the miss ratio for LRU is smaller than or equal to the miss ratio for FIFO.
The Annals of Probability, 1999
We study occurrence and properties of percolation of occupied bonds in systems with random intera... more We study occurrence and properties of percolation of occupied bonds in systems with random interactions and, hence, frustration. We develop a general argument, somewhat like Peierls' argument, by which we show that in Z d , d ≥ 2, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter p ∈ 0 1 , regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of p for which percolation occurs for all, almost all but not all, almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems.
The Annals of Probability, 1988
The Annals of Applied Probability, 1995
In this paper we present an asymptotic estimator, obtained by observing a noisy image, for the pa... more In this paper we present an asymptotic estimator, obtained by observing a noisy image, for the parameters of both a stationary Markov random field and an independent Bernoulli noise. We first estimate the parameter of the noise, by solving a polynomial equation of moderate degree (about 6-7 in the one-dimensional Ising model, and about 10-15 in the two-dimensional Ising model, for instance), and then apply the maximum pseudolikelihood method after removing the noise. Our method requires no extra simulation, and is likely to be applicable to any Markov random field, in any dimension. Here, we present the general theory and some examples in one dimension; more interesting examples in two dimensions will be discussed at length in a companion paper.
Abstract We consider the classic problem of estimating the total number T of species in a populat... more Abstract We consider the classic problem of estimating the total number T of species in a population, from repeated counts in a simple random sample. We first show that one of the most widely used estimators can be obtained via a Bayesian method from a Dirichlet prior, ...
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Papers by Alberto Gandolfi