A note on the discrete Gaussian Free Field with disordered pinning on Z^d, d\geq 2

We study the discrete massless Gaussian Free Field on $\Z^d$, $d\geq2$, in the presence of a disordered square-well potential supported on a finite strip around zero. The disorder is introduced by reward/penalty interaction coefficients, which are given by i.i.d. random variables. Under minimal assumptions on the law of the environment, we prove that the quenched free energy associated to this model exists in $\R^+$, is deterministic, and strictly smaller than the annealed free energy whenever the latter is strictly positive.Comment: 17 page

Parameter estimation and treatment optimization in a stochastic model for immunotherapy of cancer

Adoptive Cell Transfer therapy of cancer is currently in full development and mathematical modeling is playing a critical role in this area. We study a stochastic model developed by Baar et al. in 2015 for modeling immunotherapy against melanoma skin cancer. First, we estimate the parameters of the deterministic limit of the model based on biological data of tumor growth in mice. A Nonlinear Mixed Effects Model is estimated by the Stochastic Approximation Expectation Maximization algorithm. With the estimated parameters, we head back to the stochastic model and calculate the probability that the T cells all get exhausted during the treatment. We show that for some relevant parameter values, an early relapse is due to stochastic fluctuations (complete T cells exhaustion) with a non negligible probability. Then, focusing on the relapse related to the T cell exhaustion, we propose to optimize the treatment plan (treatment doses and restimulation times) by minimizing the T cell exhaustion probability in the parameter estimation ranges.Comment: major reorganisation of the paper and the reformulation of many substantial part

Crossing a fitness valley as a metastable transition in a stochastic population model

We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0,1,...,L} and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is random, exponentially distributed.Comment: Second round of revision. 40 pages, 4 figure

A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature $${\beta\geq 0}$$ are of the form $${\alpha\mu^{+}_\beta + (1-\alpha)\mu^{-}_\beta}$$ , where $${\mu^{+}_\beta}$$ and $${\mu^{-}_\beta}$$ are the two pure phases and $${0\leq\alpha\leq 1}$$ . We present here a new approach to this result, with a number of advantages: (a) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (b) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (c) this new approach might be applicable to systems for which the classical method fail

Absence of Dobrushin states for 2d2d long-range Ising models

We consider the two-dimensional Ising model with long-range pair interactions of the form $J_{xy}\sim|x-y|^{-\alpha}$ with $\alpha>2$, mostly when $J_{xy} \geq 0$. We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed $\pm$-boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman-Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts.Comment: revised versio

Stochastic individual-based models with power law mutation rate on a general finite trait space

We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph $G=(V,E)$. The evolution of the population is driven by births, deaths, competition, and mutations along the edges of $G$. We are interested in the large population limit under a mutation rate $\mu_K$ given by a negative power of the carrying capacity $K$ of the system: $\mu_K=K^{-1/\alpha},\alpha>0$. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the $\log K$ time scale, as $K$ tends to infinity. Using techniques developed in [Champagnat, M\'el\'eard, Tran, 2019] we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work generalises [Kraut, Bovier, 2019] to the stochastic setting, and Theorem 3.2 of [Bovier, Coquille, Smadi, 2018] to any finite mutation graph. We illustrate our theorem by a series of examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations.Comment: 39 pages, 7 figure

Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees

We consider $\mathbb Z$-valued $p$-SOS-models with nearest neighbor interactions of the form $|\omega_v-\omega_w|^p$, and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any symmetries of the tree. This generalises the results of Gandolfo, Ruiz and Shlosman \cite{GRS12} about the Ising model, and shows that the latter behaviour is robust. We treat three different types of extensions: non-compact state space gradient models, models without spin-symmetry, and models in small random fields. We give a detailed construction and full proofs of the extremality of the low-temperature states in the set of all Gibbs measures, analysing excess energies relative to the ground states, convergence of low-temperature expansions, and properties of cutsets.Comment: 27 pages, 3 figure

On the Gibbs states of the noncritical Potts model on Z2\mathbb Z ^2

We prove that all Gibbs states of the $$q$$ -state nearest neighbor Potts model on $$\mathbb Z ^2$$ below the critical temperature are convex combinations of the $$q$$ pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature $$\beta >\beta _c (q) = \log \left(1+\sqrt{q}\right)$$