Antonio Auffinger
YOU?
Author Swipe
View article: The Spherical $$p+s$$ Spin Glass at Zero Temperature
The Spherical $$p+s$$ Spin Glass at Zero Temperature Open
We determine the structure of the Parisi measure at zero temperature for the spherical $$p+s$$ spin glass model. We show that depending on the values of p and s , four scenarios may emerge, including the existence of 1-FRSB and 2-FRSB…
View article: On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses
On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses Open
We show that in mean-field spin glasses, a discontinuous breaking of replica symmetry at the critical inverse temperature $β_c$ implies the existence of an intermediate shattered phase. This confirms a prediction from physics regarding the…
View article: On the time constant of high dimensional first passage percolation, revisited
On the time constant of high dimensional first passage percolation, revisited Open
In [2], it was claimed that the time constant $μ_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $μ_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(τ_{e})_{e\in \mathbb E^{d}}$ are i.i.d., w…
View article: Equilibrium Distributions for t-distributed Stochastic Neighbour Embedding
Equilibrium Distributions for t-distributed Stochastic Neighbour Embedding Open
We study the empirical measure of the output of the t-distributed stochastic neighbour embedding algorithm when the initial data is given by n independent, identically distributed inputs. We prove that under certain assumptions on the dist…
View article: The Spherical p+s Spin Glass At Zero Temperature
The Spherical p+s Spin Glass At Zero Temperature Open
We determine the structure of the Parisi measure at zero temperature for the spherical p+s spin glass model. We show that depending on the values of p and s, four scenarios may emerge, including the existence of 1-FRSB and 2-FRSB phases as…
View article: Complexity of Gaussian random fields with isotropic increments: critical points with given indices
Complexity of Gaussian random fields with isotropic increments: critical points with given indices Open
We study the landscape complexity of the Hamiltonian $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is a smooth Gaussian process with isotropic increments on $\mathbb R^{N}$. This model describes a single particle on a random potential in stat…
View article: Optimization of random high-dimensional functions: Structure and algorithms
Optimization of random high-dimensional functions: Structure and algorithms Open
Replica symmetry breaking postulates that near optima of spin glass Hamiltonians have an ultrametric structure. Namely, near optima can be associated to leaves of a tree, and the Euclidean distance between them corresponds to the distance …
View article: Sharp complexity asymptotics and topological trivialization for the (<i>p</i>, <i>k</i>) spiked tensor model
Sharp complexity asymptotics and topological trivialization for the (<i>p</i>, <i>k</i>) spiked tensor model Open
Using precise random matrix theory tools and the Kac–Rice formula, we provide sharp O(1) asymptotics for the average number of deep minima of the (p, k) spiked tensor model. These sharp estimates allow us to prove that, when the signal-to-…
View article: Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups
Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups Open
We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given…
View article: Thouless-Anderson-Palmer equations for the Ghatak-Sherrington mean field spin glass model
Thouless-Anderson-Palmer equations for the Ghatak-Sherrington mean field spin glass model Open
We derive the Thouless-Anderson-Palmer (TAP) equations for the Ghatak and Sherrington model. Our derivation, based on the cavity method, holds at high temperature and at all values of the crystal field. It confirms the prediction of Yokota.
View article: The number of saddles of the spherical $p$-spin model
The number of saddles of the spherical $p$-spin model Open
We show that the quenched complexity of saddles of the spherical pure $p$-spin model agrees with the annealed complexity when both are positive. Precisely, we show that the second moment of the number of critical values of a given finite i…
View article: Complexity of Gaussian random fields with isotropic increments
Complexity of Gaussian random fields with isotropic increments Open
We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of level sets of smooth random fields on $\mathbb R^{N}$ of the form $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is a …
View article: Complexity of high dimensional gaussian random fields with isotropic increments
Complexity of high dimensional gaussian random fields with isotropic increments Open
We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of sublevel sets of smooth random fields on $\mathbb R^{N}$ of the form $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is…
View article: On properties of the spherical mixed vector p-spin model
On properties of the spherical mixed vector p-spin model Open
This paper studies properties of the mixed spherical vector p-spin model. At zero temperature, we establish and investigate a Parisi type formula for the ground state energy. At finite temperature, we provide some properties of minimizers …
View article: Topologies of Random Geometric Complexes on Riemannian Manifolds in the Thermodynamic Limit
Topologies of Random Geometric Complexes on Riemannian Manifolds in the Thermodynamic Limit Open
We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called “thermodynamic” regime. We prove the existence of universal limit laws for the topologies; namely, the ra…
View article: Topologies of random geometric complexes on Riemannian manifolds in the\n thermodynamic limit
Topologies of random geometric complexes on Riemannian manifolds in the\n thermodynamic limit Open
We investigate the topologies of random geometric complexes built over random\npoints sampled on Riemannian manifolds in the so-called "thermodynamic" regime.\nWe prove the existence of universal limit laws for the topologies; namely, the\…
View article: On concentration properties of disordered Hamiltonians
On concentration properties of disordered Hamiltonians Open
We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy $F$ with respect to the inverse temperature $\beta$, we show that the Hamiltonian concentrates around the…
View article: 50 Years of First-Passage Percolation
50 Years of First-Passage Percolation Open
We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 yea…
View article: Pemantle's min-plus binary tree
Pemantle's min-plus binary tree Open
We consider a stochastic process that describes several particles interacting by either merging or annihilation. When two particles merge, they combine their masses; when annihilation occurs, only the particle of smallest mass survives. Pa…
View article: On concentration properties of disordered Hamiltonians
On concentration properties of disordered Hamiltonians Open
We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy $F$ with respect to the inverse temperature $β$, we show that the Hamiltonian concentrates around the ene…
View article: The SK model is infinite step replica symmetry breaking at zero temperature
The SK model is infinite step replica symmetry breaking at zero temperature Open
We prove that the Parisi measure of the mixed p-spin model at zero temperature has infinitely many points in its support. This establishes Parisi's prediction that the functional order parameter of the Sherrington-Kirkpatrick model is not …
View article: On the energy landscape of spherical spin glasses
On the energy landscape of spherical spin glasses Open
We investigate the energy landscape of the spherical mixed even p-spin model near its maximum energy. We relate the distance between pairs of near maxima to the support of the Parisi measure at zero temperature. We then provide an algebrai…
View article: A duality principle in spin glasses
A duality principle in spin glasses Open
We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature para…
View article: Thouless-Anderson-Palmer equations for conditional Gibbs measures in the generic p-spin glass model
Thouless-Anderson-Palmer equations for conditional Gibbs measures in the generic p-spin glass model Open
We study the Thouless-Anderson-Palmer (TAP) equations for spin glasses on the hypercube. First, using a random, approximately ultrametric decomposition of the hypercube, we decompose the Gibbs measure, $\langle\cdot\rangle_N$, into a mixtu…
View article: Parisi formula for the ground state energy in the mixed p-spin model
Parisi formula for the ground state energy in the mixed p-spin model Open
We show that the thermodynamic limit of the ground state energy in the mixed p-spin model can be identified as a variational problem. This gives a natural generalization of the Parisi formula at zero temperature.
View article: A duality principle in spin glasses
A duality principle in spin glasses Open
We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature para…