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Pietro Caputo

Publications and source records attributed to Pietro Caputo.

72 records · Page 4Linked to original sources
arXiv2010

Polymer dynamics in the depinned phase: metastability with logarithmic barriers

We consider the stochastic evolution of a (1 + 1)-dimensional polymer in the depinned regime. At equilibrium the system exhibits a double well structure: the polymer lies(essentially) either above or below the repulsive line. As a consequence one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However the energy barrier between these two phases is only logarithmic in the system size L and therefore the two relevant time scales are only polynomial in L with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. In particular the usual scenario in which the tunneling time coincides with the exponential of the energy barrier breaks down. Our main results are: (i) a proof that the mixing time of the system lies between L^{5/2} and L^{5/2+2}; (ii) the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap.

math.PR↗
arXiv2010

Circular Law Theorem for Random Markov Matrices

Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.

math.PR↗
arXiv2010

Spectrum of large random reversible Markov chains: two examples

We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior at the edge, including the so called spectral gap. Results are obtained for two simple models with distinct limiting features. The first model is built on the complete graph while the second is a birth-and-death dynamics. Both models give rise to random matrices with non independent entries.

math.PR↗
arXiv2010

Convergence to equilibrium of biased plane partitions

We study a single-flip dynamics for the monotone surface in (2+1) dimensions obtained from a boxed plane partition. The surface is analyzed as a system of non-intersecting simple paths. When the flips have a non-zero bias we prove that there is a positive spectral gap uniformly in the boundary conditions and in the size of the system. Under the same assumptions, for a system of size M, the mixing time is shown to be of order M up to logarithmic corrections.

math.PR↗
arXiv2009

Proof of Aldous' spectral gap conjecture

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.

math.PR↗
arXiv2008

On the spectral gap of the Kac walk and other binary collision processes

We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general observation which allows to reduce the analysis of the spectral gap of an N-component system to that of the same system for N=3. The method applies to a number of random 'binary collision' processes with complete-graph structure, including non-homogeneous examples such as exclusion and colored exclusion processes with site disorder.

math.PR↗
arXiv2008

On the approach to equilibrium for a polymer with adsorption and repulsion

We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path ηhas a weight λ^{N(η)}, where λ>0 and N(η) is the number of zeros in η. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength λis varied. In this paper we study a natural 'spin flip' dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (λ=1, no wall), where the gap and the mixing time are known to scale as L^{-2} and L^2\log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for λ\geq 1 the relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (λ< 1) the gap is shown to be O(L^{-5/2}), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.

math.PR↗
arXiv2007

Convex Entropy Decay via the Bochner-Bakry-Emery approach

We develop a method, based on a Bochner-type identity, to obtain estimates on the exponential rate of decay of the relative entropy from equilibrium of Markov processes in discrete settings. When this method applies the relative entropy decays in a convex way. The method is shown to be rather powerful when applied to a class of birth and death processes. We then consider other examples, including inhomogeneous zero-range processes and Bernoulli-Laplace models. For these two models, known results were limited to the homogeneous case, and obtained via the martingale approach, whose applicability to inhomogeneous models is still unclear.

math.PR↗
arXiv2007

Isoperimetric inequalities and mixing time for a random walk on a random point process

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $α$-power of jump length. The case $α=1$ corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for $α\in(0,d)$, that the random walk confined to a cubic box of side $L$ has a.s. Cheeger constant of order at least $L^{-1}$ and mixing time of order $L^2$. For the Poisson point process, we prove that at $α=d$, there is a transition from diffusive to subdiffusive behavior of the mixing time.

math.PR↗
arXiv2007

Analysis of top-swap shuffling for genome rearrangements

We study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied in this paper is the top-swap Markov chain. The top-swap chain is a card-shuffling process with $n$ cards divided over $k$ decks, where the cards are ordered within each deck. A transition consists of choosing a random pair of cards, and if the cards lie in different decks, we cut each deck at the chosen card and exchange the tops of the two decks. We prove precise bounds on the relaxation time (inverse spectral gap) of the top-swap chain. In particular, we prove the relaxation time is $Θ(n+k)$. This resolves an open question of Durrett.

math.PR↗
arXiv2006

Entropy dissipation estimates in a Zero-Range dynamics

We study the exponential decay of relative entropy functionals for zero-range processes on the complete graph. For the standard model with rates increasing at infinity we prove entropy dissipation estimates, uniformly over the number of particles and the number of vertices.

math.PR↗
arXiv2005

Spectral gap estimates for interacting particle systems via a Bochner-type identity

We develop a general technique, based on a Bochner-type identity, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained.

math.PR↗
arXiv2004

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup $P_t$. A fundamental and still largely open problem is the understanding of the long time behavior of $\d_\h P_t$ when the initial configuration $\h$ is sampled from a highly disordered state $ν$ (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular $b$-ary tree $\Tree^b$, we tackle the above problem for the Ising and hard core gas (independent sets) models on $\Tree^b$. If $ν$ is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove $ν$-almost sure weak convergence of $\d_\h P_t$ to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time $t$. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.

math.PR↗
arXiv2002

Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models

Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with $N$ particles in a rectangle of $\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at row $h\pm 1$ with rate $q^{\pm 1}$, where $q\in (0,1)$ is a measure of the drift driving the particles towards the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in $N$ and in the size of the rectangle. The proof is inspired by a recent interesting technique envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for the non linear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin--S, ${\rm S}\in \frac12\bbN$, XXZ chain and for the 111 interface of the spin--S XXZ Heisenberg model, thus generalizing previous results valid only for spin $\frac12$.

math.PR↗
arXiv2001

Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let $L$ be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the $\hat z$ axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as $L\to \infty$. In \cite{BCNS} it was proved that, by perturbing in a sub--cylinder with basis of linear size $R\ll L$ the interface ground state, it is possible to construct excited states whose energy gap shrinks as $R^{-2}$. Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from below by $\text{const.}L^{-2}$. We prove the result by first mapping the problem into an asymmetric simple exclusion process on $\Z^3$ and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in \cite{CancMart}. Along the way we improve some bounds on the equivalence of ensembles already discussed in \cite{BCNS} and we establish an upper bound on the density of states close to the bottom of the spectrum.

cond-mat.stat-mech↗
arXiv1999

A Note on Wetting Transition for Gradient Fields

We prove existence of a wetting transition for two types of gradient fields: 1) Continuous SOS models in any dimension and 2) Massless Gaussian model in two dimensions. Combined with a recent result showing the absence of such a transition for Gaussian models above two dimensions by Bolthausen et al, this shows in particular that absolute-value and quadratic interactions can give rise to completely different behaviors.

math.PR↗