# Giambattista Giacomin

Professor

Université de Paris

UFR de Mathématiques and LPSM

Bâtiment Sophie Germain

8 place Aurélie Nemours 75013 Paris

*Office* 545 (5^{th} floor): directions

*Phone* (33)-0157279317

*E-mail* giacomin*@lpsm.paris

And if you really insist on a more realistic image...

Université de Paris UFR de mahématiques LPSM local links

January 25th and 26th, 2022

June 18^{th} to 20^{th} 2018: LPSM
Kick-off Conference

October 16^{th} 2017: a day to celebrate Herbert Spohn, Honoris Causa Doctor of Paris Diderot. INTERACTING STOCHASTIC SYSTEMS - RECENT DEVELOPMENTS

Trimester IHP (April 3^{rd}-July 7^{th} 2017): STOCHASTIC DYNAMICS OUT OF EQUILIBRIUM

#### Research interests: probability and applications

- Statistical Mechanics, Disordered Systems
- Nonequilibrium Statistical Mechanics
- Applications to Biology and Life Science

Publications (books, articles, preprints,...) and some teaching material

Désordre et fluctuations dans les systèmes physiques : Giorgio Parisi et ses défis pour les mathématiciens An article for the INSMI letter (November 2021)

Disorder and critical phenomena through basic probability models Lecture Notes in Mathematics 2025, Springer, 2011.

Random Polymer Models IC Press, World Scientific, 2007.

#### Coauthors (chronological order)

##### Disorder and criticality: critical behavior for relevant disorder models

The aim is understanding the effect of disorder on phase transitions and critical phenomena. Consider a non disordered (« pure ») statistical mechanics model that undergoes a phase transition and for which we understand the critical phenomenon, i.e. how the system behaves close to the transition (example: the two dimensional Ising model). Is this behavior stable under introduction of impurities (disorder)? Renormalization group ideas lead to the notion of disorder irrelevance and disorder relevance: in the first case the critical behavior of the system is essentially unaffected by the disorder, in the second case the nature of the critical phenomenon is determined by the disorder. Intriguing ideas have been developed in the physical literature about these issues (see Disorder and critical phenomena through basic probability models for a mathematical presentation). Even leaving aside mathematical rigor, the disorder relevant case is a wide open domain. And, in spite of sharp physical conjectures, for the two dimensional Ising model with bond disorder (for example: dilution) mathematical results are lacking, in spite of the fact that (weak) disorder is expected to be irrelevant.

In Pinning and disorder relevance for the lattice Gaussian free field (joint work with H. Lacoin) we have been able to understand the effect of of disorder on the localization transition of a (hyper-)surface pinning model: it is case of disorder relevance. The free (i.e. without pinning potentials) model is the lattice free field in dimension three or more: hence the surface or interface is rigid for the free model. It is rather straightforward to see that the pinning transition in absence of disorder is of first order. As soon as a disorder is introduced the transition becomes second order. The result is proven for very general disorder distribution. When the disorder is Gaussian we show a precise result in the sense that the free energy density is shown to have a quadratic behavior, both from above and below. This is the first instance in which the critical exponent is rigorously determined for a disorder relevant model. In Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case, Hubert Lacoin has been able to deal with the two dimensional case (rough interface): disorder is also relevant, with the transition that is of infinite order when disorder is introduced, while the pure model has a second order transition. It is interesting to note that we have proven that disorder is irrelevant for the very closely related wetting model (at least for rigid interfaces: Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger)

In Disorder and critical phenomena: the α=0 copolymer model we have then been able to tackle another model. The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent 1+α≥1. The pure version of the model has a first order (localization) transition. We show that in the α=0 the (disordered) copolymer model the localization transition becomes of infinite order. Precise estimates on the free energy density near criticality are obtained.

##### On the long time effect of small noise on stochastic ODEs (in presence of attracting periodic trajectories)

Stable limit cycles are omnipresent when modeling real life systems and this just reflects the central role that rhythms play in our life and in just about everything that surrounds us. In the image, in black, the limit cycle of an Ordinary Differential Equation (ODE) system dx(t)/dt=F(x(t)): the particular instance is the Fitzhugh-Nagumo system, a basic two dimensional model of one neuron. Perturbing stochastically the system is a way of going farther from a modeling viewpoint. The figure shows two (blue and red) one period realizations of the solution to the Itô stochastic system dX(t)=F(X(t))dt+εG(X(t))dB(t), with G(·) a matrix valued function and B a (multi-dimensional) Brownian motion and ε is a small positive parameter. Initial condition is kept the same in all cases. The first remark is that for ε tending to zero stochastic and deterministic and stochastic trajectories get closer on every finite time horizon. The question is: what happens if we let the time horizon grow with ε? When does this proximity breaks down and what do we observe? In Small noise and long time phase diffusion in stochastic limit cycle oscillators (joint work with C. Poquet and A. Shapira) we attempt a review the very wide applied science literature on this important issue and we establish rigorously (and quantitatively) that a phase diffusion phenomenon takes place on the time scale ε^{-2}. One can have a glimpse on this phenomenon by clicking on the image (The blue dot is the deterministic system, the red dot is the stochastic one). It is only going on times of about 100 (ε=1/10 in the simulation) periods that one starts appreciating the *noise induced frequency shift* induced by the noise (in mathematical terms: the phase diffusion has a net drift that happens to be in the sense of rotation of the limit cycle for the specific choice of F and G). Our work provides estimates on the effect of the noise up to times of the order of exp(cε^{-2}), c>0, where Large Deviations effects enter the game.

##### The generalized Poland-Scheraga DNA denaturation model and bivariate renewals

The Poland-Scheraga (PS) is the standard basic model for DNA denaturation, that is the transition that happens at high temperature from two complementary DNA strands that are bind together (*localized state*) to two free strands (*delocalized state*). The original PS model is limited to exact complementarity of the two strands - equal length strands and no mismatches are allowed - and it boils down to a sequence of bind pairs and symmetric loops (i.e., the number of bases contributed by each strand is the same). This model enjoyed and still enjoys a large popularity also because, in its homogeneous version, it is exactly solvable (in the sense of statistical mechanics: the PS model is a Gibbs measure) and the denaturation transition can be understood in detail. This solvable character is ultimately related to the fact that the homogeneous PS model is very closely related to a class of discrete renewal processes (it can even be mapped to a renewal). Very remarkably, a natural generalization of the PS model - the generalized PS (gPS) model - allowing unequal length strands and asymmetric loops turns out to retain the solvable character of the PS model. A direct representation of a trajectory of the model with two strands of respective lengths twelve and nineteen bases is shown in the first figure. In Generalized Poland-Scheraga denaturation model and two-dimensional renewal processes (joint work with M. Khatib) we considered the homogeneous gPS model and we have exploited the fact that this model can be mapped to a two dimensional (or bivariate) renewal. The mapping at the level of trajectory transforms base pairs into points in the plane: the base pairs (1,1), (2,2), (3,6),… become an *increasing* sequence in the plane (see the second figure: we made the conventional choice to shift all coordinates down by one). This allows an analysis which is parallel to the original PS model, with the novelties introduced by the higher dimensional character. These novelties are not only technical, because the gPS has a phenomenology that is substantially richer than the PS model. The most evident novelty is the appearance of transitions inside the localized regime. From a mathematical viewpoint these transitions between *different* localized states can be interpreted as the switching of the underlying renewal (that is under the effect of pinning potentials) from Large Deviations regimes that are of Cramér type, i.e. that correspond to tilting the measure, to regimes that are not. In DNA melting structures in the generalized Poland-Scheraga model (joint work with Q. Berger and M. Khatib) we obtain sharp estimates on the partition function in the non-Cramér regime and this allows to pin down the precise behavior of the trajectories of the system and unravel the geometric richness of the model.