Searcharxiv⌕ Search

arXiv subjects

Thimothée Thiery

Publications and source records attributed to Thimothée Thiery.

16 recordsLinked to original sources
arXiv2026

Validation of methods to estimate the uncertainty of buildings energy savings in a controlled numerical setting and Bayesian energy signature with autocorrelated errors

In the field of building energy efficiency, the measurement and verification (M&V) of energy savings following energy efficiency measures often relies on the use of a calibrated statistical model. In order to obtain reliable estimates, the estimation of uncertainties associated with this procedure is recognized as a crucial aspect of M&V. Several approaches have been proposed in the literature to compute uncertainties but a recent work has raised serious doubts on their accuracy. This is especially the case for energy data with fine time resolution (hourly and daily), for which the issues associated with autocorrelations become more important. This work focuses on this issue for daily data with three main contributions: (i) a detailed comprehensive review of several methods proposed in the literature for linear and non-linear regression models (including exact formulas for linear models, the ASHRAE 14 approximate formula and resampling approaches) ; (ii) the improvement of approaches based on Bayesian modeling for non-linear regression models with autocorrelated residuals ; (iii) a large scale numerical test of the different approaches based on a synthetic dataset containing thousands of stochastic thermal dynamic simulations. Our results permit to test the accuracy of different approaches in a controlled numerical setting. In contrast with previous work, we find in particular that our Bayesian approach leads on our dataset to consistent estimates of uncertainties. This suggests that the proposed Bayesian approach offers a convincing method to estimate uncertainties in non-linear regression models with autocorrelated residuals often encountered M&V.

stat.AP↗
arXiv2019

Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions

We revisit the Lieb-Liniger model for $n$ bosons in one dimension with attractive delta interaction in a half-space $\mathbb{R}^+$ with diagonal boundary conditions. This model is integrable for arbitrary value of $b \in \mathbb{R}$, the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as $b$ is decreased from the hard-wall case $b=+\infty$, with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar-Parisi-Zhang equation for the growth of a one-dimensional interface of height $h(x,t)$, on the half-space with boundary condition $\partial_x h(x,t)|_{x=0}=b$ and droplet initial condition at the wall. We obtain explicit expressions, valid at all time $t$ and arbitrary $b$, for the integer exponential (one-point) moments of the KPZ height field $\bar{e^{n h(0,t)}}$. From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for $b>-\frac{1}{2}$ where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for $b<-\frac{1}{2}$, where the PDF is a Gaussian. At the critical point $b=-\frac{1}{2}$, the PDF is given by the GOE Tracy-Widom distribution.

cond-mat.stat-mech↗
arXiv2019

Correlations between avalanches in the depinning dynamics of elastic interfaces

We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a $ε=d_c-d$ expansion below the upper critical dimension $d_c$ of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform displacement $W$ of the interface. For (i) we obtain the correlations of the total sizes, of the local sizes and of the total sizes with given seeds (starting points). For (ii) we obtain the correlations of the velocities, of the durations, and of the avalanche shapes. In general we find that the avalanches are {\it anti-correlated}, the occurence of a larger avalanche making more likely the occurence of a smaller one, and vice-versa. Examining the universality of our results leads us to conjecture several new exact scaling relations for the critical exponents that characterize the different distributions of correlations. The avalanche size predictions are confronted to numerical simulations for a $d=1$ interface with short range elasticity. They are also compared to our recent related work on static avalanches (shocks). Finally we show that the naive extrapolation of our result into the thermally activated creep regime at finite temperature, predicts strong positive correlations between the forward motion events, as recently observed in numerical simulations.

cond-mat.dis-nn↗
arXiv2018

Many-body delocalization as a quantum avalanche

We propose a multi-scale diagonalization scheme to study disordered one-dimensional chains, in particular the transition between many-body localization (MBL) and the ergodic phase, expected to be governed by resonant spots. Our scheme focuses on the dichotomy MBL versus {ETH} (eigenstate thermalization hypothesis). We show that a few natural assumptions imply that the system is localized with probability one at criticality. On the ergodic side, delocalization is induced by a quantum avalanche seeded by large ergodic spots, whose size diverges at the transition. On the MBL side, the typical localization length tends to a finite universal value at the transition, but there is a divergent length scale related to the response to an inclusion of large ergodic spots. A mean field approximation analytically illustrates these results and predicts as a power-law distribution for thermal inclusions at criticality.

cond-mat.dis-nn↗
arXiv2017

A microscopically motivated renormalization scheme for the MBL/ETH transition

We introduce a multi-scale diagonalization scheme to study the transition between the many-body localized and the ergodic phase in disordered quantum chains. The scheme assumes a sharp dichotomy between subsystems that behave as localized and resonant spots that obey the Eigenstate Thermalization Hypothesis (ETH). We establish a set of microscopic principles defining the diagonalization scheme, and use them to numerically study the transition in very large systems. To a large extent the results are in agreement with an analytically tractable mean-field analysis of the scheme: We find that at the critical point the system is almost surely localized in the thermodynamic limit, hosting a set of thermal inclusions whose sizes are power-law distributed. On the localized side the {\em typical} localization length is bounded from above. The bound saturates upon approach to criticality, entailing that a finite ergodic inclusion thermalizes a region of diverging diameter. The dominant thermal inclusions have a fractal structure, implying that averaged correlators decay as stretched exponentials throughout the localized phase. Slightly on the ergodic side thermalization occurs through an avalanche instability of the nearly localized bulk, whereby rare, supercritically large ergodic spots eventually thermalize the entire sample. Their size diverges at the transition, while their density vanishes. The non-local, avalanche-like nature of this instability entails a breakdown of single parameter scaling and puts the delocalization transition outside the realm of standard critical phenomena.

cond-mat.stat-mech↗
arXiv2017

Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice

We study the recently introduced Inverse-Beta polymer, an exactly solvable, anisotropic finite temperature model of directed polymer on the square lattice, and obtain its stationary measure. In parallel we introduce an anisotropic zero temperature model of directed polymer on the square lattice, the Bernoulli-Geometric polymer, and obtain its stationary measure. This new exactly solvable model is dual to the Inverse-Beta polymer and interpolates between models of first and last passage percolation on the square lattice. Both stationary measures are shown to satisfy detailed balance. We also obtain the asymptotic mean value of (i) the free-energy of the Inverse-Beta polymer; (ii) the optimal energy of the Bernoulli-Geometric polymer. We discuss the convergence of both models to their stationary state. We perform simulations of the Bernoulli-Geometric polymer that confirm our results.

cond-mat.dis-nn↗
arXiv2017

Midpoint distribution of directed polymers in the stationary regime: exact result through linear response

We obtain an exact result for the midpoint probability distribution function (pdf) of the stationary continuum directed polymer, when averaged over the disorder. It is obtained by relating that pdf to the linear response of the stochastic Burgers field to some perturbation. From the symmetries of the stochastic Burgers equation we derive a fluctuation-dissipation relation so that the pdf gets given by the stationary two space-time points correlation function of the Burgers field. An analytical expression for the latter was obtained by Imamura and Sasamoto [2013], thereby rendering our result explicit. In the large length limit that implies that the pdf is nothing but the scaling function $f_{\rm KPZ}(y)$ introduced by Prähofer and Spohn [2004]. Using the KPZ-universality paradigm, we find that this function can therefore also be interpreted as the pdf of the position y of the maximum of the Airy process minus a parabola and a two-sided Brownian motion. We provide a direct numerical test of the result through simulations of the Log-Gamma polymer.

cond-mat.dis-nn↗
arXiv2017

The induced motion of a probe coupled to a bath with random resettings

We consider a probe linearly coupled to the center of mass of a nonequilibrium bath. We study the induced motion on the probe for a model where a resetting mechanism is added to an overdamped bath dynamics with quadratic potentials. The fact that each bath-particle is at random times being reset to a fixed position is known for optimizing diffusive search strategies, but here stands for the nonequilibrium aspect of the bath. In the large bath scaling limit the probe is governed by an effective Langevin equation. Depending on the value of the parameters, there appear three regimes: (i) an equilibrium-like regime but with a reduced friction and an increased effective temperature; (ii) a regime where the noise felt by the probe is continuous but nonGaussian and exhibits fat-tails; (iii) a regime with a nonGaussian noise exhibiting power-law distributed jumps. The model thus represents an exactly solvable case for the origin of nonequilibrium probe dynamics.

cond-mat.stat-mech↗
arXiv2017

Analytical Methods and Field Theory for Disordered Systems

This thesis presents several aspects of the physics of disordered elastic systems and of the analytical methods used for their study. On one hand we will be interested in universal properties of avalanche processes in the statics and dynamics (at the depinning transition) of elastic interfaces of arbitrary dimension in disordered media at zero temperature. To study these questions we will use the functional renormalization group. After a review of these aspects we will more particularly present the results obtained during the thesis on (i) the spatial structure of avalanches and (ii) the correlations between avalanches. On the other hand we will be interested in static properties of directed polymers in $1+1$ dimension, and in particular in observables related to the KPZ universality class. In this context the study of exactly solvable models has recently led to important progress. After a review of these aspects we will be more particularly interested in exactly solvable models of directed polymer on the square lattice and present the results obtained during the thesis in this direction: (i) classification of Bethe ansatz exactly solvable models of directed polymer at finite temperature on the square lattice; (ii) KPZ universality for the Log-Gamma and Inverse-Beta models; (iii) KPZ universality and non-universality for the Beta model; (iv) stationary measures of the Inverse-Beta model and of related zero temperature models.

cond-mat.dis-nn↗
arXiv2017

Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Although time-dependent random media with short range correlations lead to (possibly biased) normal tracer diffusion, anomalous fluctuations occur away from the most probable direction. This was pointed out recently in 1D lattice random walks, where statistics related to the 1D Kardar- Parisi-Zhang (KPZ) universality class, i.e. the GUE Tracy Widom distribution, were shown to arise. Here we provide a simple picture for this correspondence, directly in the continuum as well as for lattice models, which allows to study arbitrary space dimension and to predict a variety of universal distributions. In $d = 1$ we predict and verify numerically the emergence of the GOE Tracy-Widom distribution for the fluctuations of the transition probability. In $d = 3$ we predict a phase transition from Gaussian fluctuations to 3D-KPZ type fluctuations as the bias is increased. We predict KPZ universal distributions for the arrival time of a first particle from a cloud diffusing in such media.

cond-mat.dis-nn↗
arXiv2016

Universal correlations between shocks in the ground state of elastic interfaces in disordered media

The ground state of an elastic interface in a disordered medium undergoes collective jumps upon variation of external parameters. These mesoscopic jumps are called shocks, or static avalanches. Submitting the interface to a parabolic potential centered at $w$, we study the avalanches which occur as $w$ is varied. We are interested in the correlations between the avalanche sizes $S_1$ and $S_2$ occurring at positions $w_1$ and $w_2$. Using the Functional Renormalization Group (FRG), we show that correlations exist for realistic interface models below their upper critical dimension. Notably, the connected moment $ \langle S_1 S_2 \rangle^c$ is up to a prefactor exactly the renormalized disorder correlator, itself a function of $|w_2-w_1|$. The latter is the universal function at the center of the FRG; hence correlations between shocks are universal as well. All moments and the full joint probability distribution are computed to first non-trivial order in an $ε$-expansion below the upper critical dimension. To quantify the local nature of the coupling between avalanches, we calculate the correlations of their local jumps. We finally test our predictions against simulations of a particle in random-bond and random-force disorder, with surprisingly good agreement.

cond-mat.dis-nn↗
arXiv2016

Universality in the mean spatial shape of avalanches

Quantifying the universality of avalanche observables beyond critical exponents is of current great interest in theory and experiments. Here, we improve the characterization of the spatio-temporal process inside avalanches in the universality class of the depinning of elastic interfaces in random media. Surprisingly, at variance with the temporal shape, the spatial shape of avalanches has not yet been predicted. In part this is due to a lack of an analytically tractable definition: how should the shapes be centered? Here we introduce such a definition, accessible in experiments, and study the mean spatial shape of avalanches at fixed size centered around their starting point (seed). We calculate the associated universal scaling functions, both in a mean-field model and beyond. Notably, they are predicted to exhibit a cusp singularity near the seed. The results are in good agreement with a numerical simulation of an elastic line.

cond-mat.dis-nn↗
arXiv2016

On integrable directed polymer models on the square lattice

In a recent work Povolotsky provided a three-parameter family of stochastic particle systems with zero-range interactions in one dimension which are integrable by coordinate Bethe ansatz. Using these results we obtain the corresponding condition for integrability of a class of directed polymer models with random weights on the square lattice. Analyzing the solutions we find, besides known cases, a new two-parameter family of integrable DP model, which we call the Inverse-Beta polymer, and provide its Bethe ansatz solution.

cond-mat.dis-nn↗
arXiv2015

Spatial shape of avalanches in the Brownian force model

We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.

cond-mat.dis-nn↗
arXiv2014

Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz

We study the model of a discrete directed polymer (DP) on the square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen. The integer moments of the partition sum, $\overline{Z^n}$, are studied using a transfer matrix formulation, which appears as a generalization of the Lieb-Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of $\overline{Z^n}$ for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments, and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin-Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of $Z$ and its Laplace transform. In the limit of very long DP, this ansatz yields that the distribution of the free energy converges to the GUE Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al. based on a formula obtained by Seppalainen using the gRSK correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at large time. We argue the equivalence between our formula and the one of Borodin et al. As we discuss, this open the way to explore the connections between quantum integrability and tropical geometry.

cond-mat.dis-nn↗