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Antonio Auffinger

Publications and source records attributed to Antonio Auffinger.

At least 19 recordsLinked to original sources
arXiv2025

On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses

We show that in mean-field spin glasses, a discontinuous breaking of replica symmetry at the critical inverse temperature $\beta_c$ implies the existence of an intermediate shattered phase. This confirms a prediction from physics regarding the nature of random first order phase transitions. On the other hand, we give an example of a spherical spin glass which exhibits shattering, yet the transition is continuous at $\beta_c$.

math.PR↗
arXiv2025

On the time constant of high dimensional first passage percolation, revisited

In [2], it was claimed that the time constant $\mu_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $\mu_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(\tau_{e})_{e\in \mathbb E^{d}}$ are i.i.d., with a common c.d.f. $F$ satisfying $\left|\frac{F(x)}{x}-a\right| \le \frac{C}{|\log x|}$ for some constants $a, C$ and sufficiently small $x$. However, the proof of the upper bound, namely, Equation (2.1) in [2] \begin{align} \limsup_{d\to\infty} \frac{\mu_{d}(e_{1})ad}{\log d} \le \frac{1}{2} \end{align} is incorrect. In this article, we provide a different approach that establishes this inequality. As a side product of this new method, we also show that the variance of the non-backtracking passage time to the first hyperplane is of order $o\big((\log d/d)^{2}\big)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed.

math.PR↗
arXiv2023

Equilibrium Distributions for t-distributed Stochastic Neighbour Embedding

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 distribution of the inputs, this sequence of measures converges to an equilibrium distribution, which is described as a solution of a variational problem.

math.PR↗
arXiv2022

The Spherical p+s Spin Glass At Zero Temperature

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 predicted by Crisanti and Leuzzi [10, 11]. We also provide consequences for the model at low temperature.

math.PR↗
arXiv2022

Complexity of Gaussian random fields with isotropic increments

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 Gaussian process with isotropic increments. We derive asymptotic formulas for the mean number of critical points with critical values in an open set as the dimension $N$ goes to infinity. In a companion paper, we provide the same analysis for the number of critical points with a given index.

math.PR↗
arXiv2022

Complexity of Gaussian random fields with isotropic increments: critical points with given indices

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 statistical physics. We derive asymptotic formulas for the mean number of critical points of index $k$ with critical values in an open set as the dimension $N$ goes to infinity. In a companion paper, we provide the same analysis without the index constraint.

math.PR↗
arXiv2022

Optimization of random high-dimensional functions: Structure and algorithms

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 along this tree. We survey recent progress towards a rigorous proof of this picture in the context of mixed $p$-spin spin glass models. We focus in particular on the following topics: $(i)$~The structure of critical points of the Hamiltonian; $(ii)$~The realization of the ultrametric tree as near optima of a suitable TAP free energy; $(iii)$~The construction of efficient optimization algorithm that exploits this picture.

math.PR↗
arXiv2021

Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

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 by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist.

math.PR↗
arXiv2021

The SK model is infinite step replica symmetry breaking at zero temperature

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 a step function at zero temperature. As a consequence, we show that the number of levels of broken replica symmetry in the Parisi formula of the free energy diverges as the temperature goes to zero.

math.PR↗
arXiv2021

Sharp complexity asymptotics and topological trivialization for the (p, k) spiked tensor model

We provide O(1) asymptotics for the average number of deep minima of the (p,k) spiked tensor model. We also derive an explicit formula for the limiting ground state energy on the N-dimensional sphere, similar to the work of Jagannath-Lopatto-Miolane. Moreover, when the signal to noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and we determine its limit as the signal-to-noise ratio diverges.

math.PR↗
arXiv2020

Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limit

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 random normalized counting measure of connected components (counted according to homotopy type) is shown to converge in probability to a deterministic probability measure. Moreover, we show that the support of the deterministic limiting measure equals the set of all homotopy types for Euclidean geometric complexes of the same dimension as the manifold.

math.PR↗
arXiv2020

The number of saddles of the spherical $p$-spin model

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 index in a given interval has twice the growth rate of the first moment.

math.PR↗
arXiv2020

On properties of the spherical mixed vector p-spin model

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 of the Crisanti-Sommers formula recently obtained by Justin Ko. In particular, we extend some of the one-dimensional Parisi measure results of Auffinger-Chen to the vector case.

math.PR↗
arXiv2018

Existence of two-step replica symmetry breaking for the spherical mixed p-spin glass at zero temperature

We provide the first examples of two-step replica symmetry breaking (2-RSB) models for the spherical mixed p-spin glass at zero temperature. Precisely, we show that for a certain class of mixtures, the Parisi measure at zero temperature is purely atomic and has exactly three distinct points in its support. We then derive a few consequences for the topology of the random landscape in these cases. Our main result also provides a negative answer to a question raised in 2011 by Auffinger and Ben Arous about the classification of pure-like and full mixture models.

math.PR↗
arXiv2018

On spin distributions for generic p-spin models

We provide an alternative formula for spin distributions of generic p-spin glass models. As a main application of this expression, we write spin statistics as solutions of partial differential equations and we show that the generic p-spin models satisfy multiscale Thouless-Anderson-Palmer equations as originally predicted in the work of Mezard-Virasoro.

math.PR↗
arXiv2018

Thouless-Anderson-Palmer equations for the generic p-spin glass model

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 mixture of conditional laws, $\langle\cdot\rangle_{α,N}$. We show that the TAP equations hold for the spin at any site with respect to $\langle\cdot\rangle_{α,N}$ simultaneously for all $α$. This result holds for generic models provided that the Parisi measure of the model has a jump at the top of its support.

math.PR↗
arXiv2017

Pemantle's min-plus binary tree

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. Particles start at the bottom of a binary tree of depth N and move towards the root. Assuming that merging or annihilation happens independently at random, we determine the limit law of the final mass of the system in the large N limit.

math.PR↗