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Francesco Caravenna

Publications and source records attributed to Francesco Caravenna.

At least 37 records · Page 2Linked to original sources
arXiv2018

Diameter in ultra-small scale-free random graphs: Extended version

It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately $k^{-(τ-1)}$ with $τ\in(2,3)$, typical distances between pairs of vertices in a graph of size $n$ are asymptotic to $\frac{2\log\log n}{|\log(τ-2)|}$ and $\frac{4\log\log n}{|\log(τ-2)|}$, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order $\log\log n$ precisely when the minimal forward degree $d$ of vertices is at least $2$. We identify the exact constant, which equals that of the typical distances plus $2/\log d$. Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

math.PR↗
arXiv2017

The asymptotic smile of a multiscaling stochastic volatility model

We consider a stochastic volatility model which captures relevant stylized facts of financial series, including the multi-scaling of moments. The volatility evolves according to a generalized Ornstein-Uhlenbeck processes with super-linear mean reversion. Using large deviations techniques, we determine the asymptotic shape of the implied volatility surface in any regime of small maturity $t \to 0$ or extreme log-strike $|κ| \to \infty$ (with bounded maturity). Even if the price has continuous paths, out-of-the-money implied volatility diverges for small maturity, producing a very pronounced smile.

math.PR↗
arXiv2017

Universality in marginally relevant disordered systems

We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension (1+1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated Fourth Moment Theorem, reveals an interesting chaos structure shared by all models in the above class.

math.PR↗
arXiv2016

The strong renewal theorem

We consider real random walks with positive increments (renewal processes) in the domain of attraction of a stable law with index $α\in (0,1)$. The famous local renewal theorem of Garsia and Lamperti, also called strong renewal theorem, is known to hold in complete generality only for $α> \frac{1}{2}$. Understanding when the strong renewal theorem holds for $α\le \frac{1}{2}$ is a long-standing problem, with sufficient conditions given by Williamson, Doney and Chi. In this paper we give a complete solution, providing explicit necessary and sufficient conditions (an analogous result has been independently and simultaneously proved by Doney in arXiv:1507.06790). We also show that these conditions fail to be sufficient if the random walk is allowed to take negative values. This paper is superseded by arXiv:1612.07635

math.PR↗
arXiv2016

Polynomial chaos and scaling limits of disordered systems

Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

math.PR↗
arXiv2016

General smile asymptotics with bounded maturity

We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [Math. Finance 19 (2009), 1-12]. We present applications to popular models, including Carr-Wu finite moment logstable model, Merton's jump diffusion model and Heston's model.

q-fin.PR↗
arXiv2015

Universality for the pinning model in the weak coupling regime

We consider disordered pinning models, when the return time distribution of the underlying renewal process has a polynomial tail with exponent $α\in (1/2,1)$. This corresponds to a regime where disorder is known to be relevant, i.e. to change the critical exponent of the localization transition and to induce a non-trivial shift of the critical point. We show that the free energy and critical curve have an explicit universal asymptotic behavior in the weak coupling regime, depending only on the tail of the return time distribution and not on finer details of the models. This is obtained comparing the partition functions with corresponding continuum quantities, through coarse-graining techniques.

math.PR↗
arXiv2014

The continuum disordered pinning model

Any renewal processes on $\mathbb{N}$ with a polynomial tail, with exponent $α\in (0,1)$, has a non-trivial scaling limit, known as the $α$-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for $α\in (1/2, 1)$ these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of $\mathbb{R}$ in a white noise random environment, with subtle features: -Any fixed a.s. property of the $α$-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment. -Nonetheless, the law of the CDPM is singular with respect to the law of the $α$-stable regenerative set, for almost every realization of the environment. The existence of a disordered continuum model, such as the CDPM, is a manifestation of disorder relevance for pinning models with $α\in (1/2, 1)$.

math.PR↗
arXiv2013

The Critical Curve of the Random Pinning and Copolymer Models at Weak Coupling

We study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak coupling regime, showing that it is universal. This proves a conjecture of Bolthausen, den Hollander and Opoku for copolymer models (ref. [8]), which we also extend to pinning models.

math.PR↗
arXiv2013

A general smoothing inequality for disordered polymers

This note sharpens the smoothing inequality of Giacomin and Toninelli for disordered polymers. This inequality is shown to be valid for any disorder distribution with locally finite exponential moments, and to provide an asymptotically sharp constant for weak disorder. A key tool in the proof is an estimate that compares the effect on the free energy of tilting, respectively, shifting the disorder distribution. This estimate holds in large generality (way beyond disordered polymers) and is of independent interest.

math.PR↗
arXiv2012

An invariance principle for random walk bridges conditioned to stay positive

We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits a suitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and Wachtel [38] and Doney [21]. We review and extend these relations to the absolutely continuous setting.

math.PR↗
arXiv2012

A note on directly Riemann integrable functions

A non-negative function f, defined on the real line or on a half-line, is said to be directly Riemann integrable (d.R.i.) if the upper and lower Riemann sums of f over the whole (unbounded) domain converge to the same finite limit, as the mesh of the partition vanishes. In this note we show that, for a Lebesgue-integrable function f, very mild conditions are enough to ensure that some n-fold convolution of f with itself is d.R.i.. Applications to renewal theory and to local limit theorems are discussed.

math.PR↗
arXiv2012

Lectures on random polymers

These lecture notes are a guided tour through the fascinating world of polymer chains interacting with themselves and/or with their environment. The focus is on the mathematical description of a number of physical and chemical phenomena, with particular emphasis on phase transitions and space-time scaling. The topics covered, though only a selection, are typical for the area. Sections 1-3 describe models of polymers without disorder, Sections 4-6 models of polymers with disorder. Appendices A-E contain tutorials in which a number of key techniques are explained in more detail.

math.PR↗
arXiv2012

Scaling and multiscaling in financial series: a simple model

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with a remarkable accuracy.

q-fin.ST↗
arXiv2010

The discrete-time parabolic Anderson model with heavy-tailed potential

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model.

math.PR↗
arXiv2010

The weak coupling limit of disordered copolymer models

A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena when the polymer fluctuates in a nonhomogeneous medium, for example, made of two solvents separated by an interface. One may observe, for instance, the localization of the polymer at the interface between the two solvents. A discrete model of such system, based on the simple symmetric random walk on $\mathbb {Z}$, has been investigated in [Bolthausen and den Hollander, Ann. Probab. 25 (1997), 1334-1366], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, has been established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this work, we prove that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models, obtaining as limits a one-parameter [$α\in(0,1)$] family of continuum models, based on $α$-stable regenerative sets.

math.PR↗
arXiv2010

Copolymers at selective interfaces: settled issues and open problems

We review the literature on the localization transition for the class of polymers with random potentials that goes under the name of copolymers near selective interfaces. We outline the results, sketch some of the proofs and point out the open problems in the field. We also present in detail some alternative proofs that simplify what one can find in the literature.

math.PR↗