# Roberto FernandezUtrecht University | UU · Department of Mathematics

Roberto Fernandez

## About

75

Publications

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2,424

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Citations since 2017

## Publications

Publications (75)

We study the effects of stochastic thermal fluctuations on the instability of the free surface of a flat liquid metallic film on a solid substrate. These fluctuations are represented by a stochastic noise term added to the deterministic equation for the film thickness within the long-wave approximation. Unlike the case of polymeric films, we find t...

For a general class of gas models ---which includes discrete and continuous
Gibbsian models as well as contour or polymer ensembles--- we determine a
\emph{diluteness condition} that implies: (1) Uniqueness of the infinite-volume
equilibrium measure; (2) stability of this measure under perturbations of
parameters and discretization schemes, and (3)...

We study the effects of stochastic thermal fluctuations on the instability of the free surface of a
flat liquid film upon a solid substrate. These fluctuations are represented as a standard Brownian
motion that can be added to the deterministic equation for the film thickness within the lubrication
approximation. Here, we consider that while the no...

We study the asymptotic hitting time $\tau^{(n)}$ of a family of Markov
processes $X^{(n)}$ to a target set $G^{(n)}$ when the process starts from a
trap defined by very general properties.
We give an explicit description of the law of $X^{(n)}$ conditioned to stay
within the trap, and from this we deduce the exponential distribution of
$\tau^{(n)}...

We study the hitting times of Markov processes to target set $G$, starting
from a reference configuration $x_0$ or its basin of attraction. The
configuration $x_0$ can correspond to the bottom of a (meta)stable well, while
the target $G$ could be either a set of saddle (exit) points of the well, or a
set of further (meta)stable configurations. Thre...

We continue our study of Gibbs-non-Gibbs dynamical transitions. In the
present paper we consider a system of Ising spins on a large discrete torus
with a Kac-type interaction subject to an independent spin-flip dynamics
(infinite-temperature Glauber dynamics). We show that, in accordance with the
program outlined in \cite{vEFedHoRe10}, in the therm...

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics (“infinite-temperature” dynamics). We show that, in this setup, the program outlined in van Enter et al. (Moscow Math J 10:687–711, 2010) can be fully completed, namely, Gibbs-non-Gibbs transitions are equivalent to bifurca...

Regular $g$-measures are discrete-time processes determined by conditional
expectations with respect to the past. One-dimensional Gibbs measures, on the
other hand, are fields determined by simultaneous conditioning on past and
future. For the Markovian and exponentially continuous cases both theories are
known to be equivalent. Its equivalence for...

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the sp...

We introduce jump processes in R k , called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on th...

We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cutoff phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially di...

We present a general framework linking cut-off and exit excursions for birth-and-death processes on a countable alphabet. Under suitable hypotheses, we prove that cut-off convergence towards a (local) equilibrium is accompanied by exponentially distributed out-of-equilibrium excursions. Furthermore, atypical trajectories leading to these excursions...

We establish a one-to-one correspondence between one-sided and two-sided regular systems of conditional probabilities on the half-line that preserves the associated chains and Gibbs measures. As an application, we determine uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic Ising models with long range interactions. Our st...

Motivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let \alpha be a nonnegative matrix over a finite or countably infinite index set X, and define the "cleaning operators" \beta_h = I_{1-h} + I_h \alpha for h: X \to [0,1] (here I_f denotes the diagonal matrix with entries f). We ask: For whic...

The chapter presents an introduction of the main concepts and notions of Gibbsianness and Non-Gibbsianness. Kozlov's theorem is a main tool used to detect non-Gibbsianness, which leads to an early presentation of the different non-Gibbsianness classification schemes. The chapter reviews examples of non-Gibbsianness. These examples show up through v...

We use mathematically rigorous perturbation theory to study the transition between the Mott insulator and the conjectured Bose-Einstein condensate in a hard-core Bose-Hubbard model. The critical line is established to lowest order in the tunneling amplitude.

We consider birth-and-death processes of objects (animals) defined in Z d having unit death rates and random birth rates. For animals with uniformly bounded diameter we establish conditions on the rate distribution under which the following holds for almost all realizations of the birth rates: (i) the process is ergodic with at worst power-law time...

We consider birth-and-death processes of objects (animals) defined in ${\bf Z}^d$ having unit death rates and random birth rates. For animals with uniformly bounded diameter we establish conditions on the rate distribution under which the following holds for almost all realizations of the birth rates: (i) the process is ergodic with at worst power-...

We state a construction theorem for specifications starting from single-site conditional probabilities (singleton part). We consider general single-site spaces and kernels that are absolutely continuous with respect to a chosen product measure (free measure). Under a natural order-consistency assumption and weak non-nullness requirements we show ex...

We review four types of results combining or relating the theories of discrete-time stochastic processes and of one-dimensional specifications. First we list some general properties of stochastic processes which are extremal among those consistent with a given set of transition probabilities. They include: triviality on the tail field, short-range...

We discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results betw...

We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail $\sigma$-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness...

We restore part of the thermodynamic formalism for some
renormalized measures that are known to be non-Gibbsian. We
determine a necessary and sufficient condition for consistency
with a specification that is quasilocal only in a fixed
direction. This condition is then applied to models with FKG
monotonicity and to models with appropriate directiona...

We consider chains whose transition probabilities depend on the whole past, with summable continuity rates. We show that
Ornstein's -distance between one such chain and its canonical Markov approximations of different orders is at worst proportional to the
continuity rate of the chain. The result generalizes previous bounds obtained by X. Bressaud...

We restore part of the thermodynamic formalism for some renormalized measures that are known to be non-Gibbsian. We first point out that a recent theory due to Pfister implies that for block-transformed measures free energies and relative entropy densities exist and are conjugate convex functionals. We then determine a necessary and sufficient cond...

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following:
(1) For all ν and μ, νS(t) i...

We present a class of random cellular automata with multiple invariant measures which are all non-Gibbsian. The automata have configuration space {0,1}^{Z^d}, with d > 1, and they are noisy versions of automata with the "eroder property". The noise is totally asymmetric in the sense that it allows random flippings of "0" into "1" but not the conver...

Coupling, renewal and perfect simulation of chains of infinite order
by
R. Fernández P. A. Ferrari and A. Galves
These notes are dedicated to Ted Harris who taught us how to construct
particle systems using random graphs, cutting and pasting pieces so that to put in evidence, in the most elementary way, the properties of the process. The purpose o...

We present a perfect simulation algorithm for stationary processes indexed by $\mathbb{Z}$, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but rand...

We present a perfect simulation algorithm for measures that are absolutely
continuous with respect to some Poisson process and can be obtained as
invariant measures of birth-and-death processes. Examples include area- and
perimeter-interacting point processes (with stochastic grains), invariant
measures of loss networks, and the Ising contour and r...

We compute the speed of convergence of the canonical Markov approximation of a chain with complete connections with summable decays. We show that in the -topology the approximation converges at least at a rate proportional to these decays. This is proven by explicitly constructing a coupling between the chain and each range-k approximation.

For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram.

I review issues related with the presence or absence of Gibbsianness in measures describing random fields in lattices. After a brief exposition of the definition and properties of Gibbs measures, I discuss the phenomenon of non-Gibbsianness, its examples, characterization, the proposed classification schemes and notions of generalized Gibbsianness....

We present rigorous results for several variants of the Hubbard model in the strong-coupling regime. We establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are, in fact, leading-order terms of well defined (volume-independent) unitarily equivalent interactions. In addition, in the...

We present an upper bound on the mixing rate of the equilibrium state of a dynamical systems defined by the one-sided shift and a non H\"{o}lder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pai...

We present a probabilistic approach for the study of systems with exclusions,
in the regime traditionally studied via cluster-expansion methods. In this
paper we focus on its application for the gases of Peierls contours found in
the study of the Ising model at low temperatures, but most of the results are
general. We realize the equilibrium measur...

We present a new approach to study measures on ensembles of contours,
polymers or other objects interacting by some sort of exclusion condition. For
concreteness we develop it here for the case of Peierls contours. Unlike
existing methods, which are based on cluster-expansion formalisms and/or
complex analysis, our method is strictly probabilistic...

We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-$\sigma$-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with mon...

We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-σ-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonici...

Starting from classical lattice systems ind2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that adding a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermi...

We show that decimation transformations applied to high-$q$ Potts models result in non-Gibbsian measures even for temperatures higher than the transition temperature. We also show that majority transformations applied to the Ising model in a very strong field at low temperatures produce non-Gibbsian measures. This shows that pathological behavior o...

Renormalization transformations were introduced in statistical mechanics to study critical points. Their natural set-up, however, is within probability theory: they are maps between probability spaces, defined by suitable probability kernels. We review several interesting questions motivated by applications in physics as well as in other areas like...

We discuss some problems which arise if one tries to implement renormalization group transformations as maps from Hamiltonians to Hamiltonians. We provide various examples, involving systems not necessarily in the vicinity of a phase transition, where this can not be done, because Gibbs measures under the action of various real-space transformation...

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). Regarding regularity, we show that the...

We show how non-Gibbsian states arise as a consequence of renormalizing Gibbs states for classical finite-spin lattice systems. We also discuss possible extensions to unbounded or quantum spins and some new problems occurring in these cases.

These lectures are concerned with the analysis and applications of functional integrals defined by small perturbations of Gaussian measures. The central topic is the renormalization-group. Following Wilson and Polchinski, an effective potential is studied as a function of an ultra-violet cutoff. By changing the cutoff in a continuous manner, one ob...

We perform an exact renormalization-group analysis of one-dimensional 4-state clock models with complex interactions. Our aim is to provide a simple explicit illustration of the behavior of the renormalization-group flow in a system exhibiting a rich phase diagram. In particular we study the flow in the vicinity of phase transitions with a first-or...

The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory are expounded in detail. The book conta...

We now consider a possibly site-dependent “magnetic field”, i.e. a vector \(h = {({h_x})_{x \in L}}\)
. In the presence of a magnetic field the random-walk expansion of S
n
(x
1,...,x
n
) undergoes a double alteration. On the one hand we must also include paths connecting one x
i
with an “external magnetic field”. This means that the walks may end...

In this chapter we study a classic problem of probability theory — the intersection properties of simple random walks — using a rigorous blend of perturbation theory and renormalization-group arguments. Aside from its intrinsic mathematical interest, this problem played a key role in motivating the field-theoretic developments described in the rema...

This chapter has been co-authored with Gerhard Hartsleben.

In this chapter we give a survey of the correlation inequalities which can be obtained using the random-walk formalism. We also discuss some profound inequalities which cannot be obtained within the random-walk formalism, but which need instead the full power of the random-current formalism. Some of the results in this chapter are given without pro...

In this chapter we describe how continuum Euclidean quantum field theories can be obtained as limits of resealed lattice theories. Our main goal is to give a precise exposition of what is known about the triviality of continuum limits for d > 4 and d = 4.

We now discuss the behavior of the weights of the previous models with regard to the partition of a family of walks into subfamilies and to the splitting of a walk. We also discuss some identities and inequalities on the derivatives of the weights with respect to J or h. Most of the results of this chapter hold for arbitrary h ≥ 0 (but the only pol...

We are now ready to set the stage for Parts II and III of this book. The method exploited in this book consists in representing spin systems in terms of interacting random walks We can then use geometrical arguments to derive properties of these random walks, which in turn imply differential inequalities for the corresponding spin systems. These de...

In this section we summarize the basic definitions and notations to be used in the sequel. Most of these have already been introduced in earlier chapters, but we repeat them here in order to make Part III as self-contained as possible. We hope that Part III will be accessible to readers interested only in the physical consequences and not in the te...

The principal goal of the theory of critical phenomena is to make quantitative predictions for universal features of critical behavior — critical exponents, universal ratios of critical amplitudes, equations of state, and so forth — as discussed in Section 1.1. (Non-universal features, such as critical temperatures, are of lesser interest.) The pre...

In this chapter1 we discuss a method for constructing continuum Euclidean field theories as scaling (continuum) limits of lattice spin systems approaching a critical point. We show what the existence of a scaling limit gives us information about the critical behavior of the lattice spin system.

In this chapter1, we describe some fairly recent results on phase transitions and critical points in classical lattice spin systems. We emphasize the analysis of explicit models and quantitative information on such models. A complementary point of view is developed in [309]. See also [447, 469].

We start by describing the different kinds of phase transitions encountered in the study of condensed matter in thermal equilibrium.

In this chapter1 we sketch a specific method for constructing scaling (≡ continuum) limits, G*(x
1,..., x
n
), of rescaled correlations, G
θ
(x
1,..., x
n
), as θ → ∞, namely the Kadanoff block spin transformations. They shall serve as a typical example of “renormalization group transformations.”

We reconsider the conceptual foundations of the renormalization-group (RG) formalism. We show that the RG map, defined on a suitable space of interactions, is always single valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the other ha...

We consider the conceptual foundations of the renormalization-group (RG) formalism. We show that the RG map, defined on a suitable space of interactions, is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the other hand...

We compare a recent result of Dobrushin and Martirosyan with previous results by Gallavotti and Miracle-Sole and by Israel and point out that the analytic behavior at high temperatures for many-particle interactions is different depending on whether the interactions are weighted with a lattice-gas or Ising norm or, on the other hand, with the supre...

A general construction of ferromagnetic systems with many phase transitions is given. It is based on two new results: an extension of one of the GKS inequalities to not necessarily ferromagnetic interactions, and a uniqueness of the Gibbs state theorem for perturbations of some simple systemsat all temperatures.

Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's [^(b)]\hat \beta
, , and 4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar pow...

For a family of translation-invariant, ferromagnetic, one-component spin systems—which includes Ising and 4 models—we prove that (i) the phase transition is sharp in the sense that at zero magnetic field the high- and low-temperature phases extend up to a common critical point, and (ii) the critical exponent obeys the mean field bound 1/2. The pres...

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind4 dimensions the magnetization is continuous atT
c and its critical exponents take the classical values=3 and=1/2, with possible logarithmic...

We introduce jump processes in Rk, called density-prole process , to model biological signaling networks. They describe the macroscopic evo- lution of nite-size spin-ip models with k types of spins interacting through a non-reversible Glauber dynamics. We focus on the the k- dimensional empirical-magnetization vector in the thermodynamic limit, and...

We consider birth-and-death processes of objects (animals) defined in $\Z^d$ having unit death rates and random birth rates. For animals with uniformly bounded diameter we establish conditions on the rate distribution under which the following holds for almost all realizations of the birth rates: (i) the process is ergodic with at worst power-law t...