arXiv subjects

C. Giardina

Publications and source records attributed to C. Giardina.

4 recordsLinked to original sources

arXiv2012

Nonconventional averages along arithmetic progressions and lattice spin systems

We study the so-called nonconventional averages in the context of lattice spin systems, or equivalently random colourings of the integers. For i.i.d. colourings, we prove a large deviation principle for the number of monochromatic arithmetic progressions of size two in the box $[1,N]\cap \N$, as $N\to\infty$, with an explicit rate function related to the one-dimensional Ising model. For more general colourings, we prove some bounds for the number of monochromatic arithmetic progressions of arbitrary size, as well as for the maximal progression inside the box $[1,N]\cap \N$. Finally, we relate nonconventional sums along arithmetic progressions of size greater than two to statistical mechanics models in dimension larger than one.

math.PR↗

arXiv2010

The inclusion process: duality and correlation inequalities

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other -- a process which we call here the symmetric inclusion process (SIP) -- or repel each other -- a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction, -- the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.

math.PR↗

arXiv2009

Matching with shift for one-dimensional Gibbs measures

We consider matching with shifts for Gibbsian sequences. We prove that the maximal overlap behaves as $c\log n$, where $c$ is explicitly identified in terms of the thermodynamic quantities (pressure) of the underlying potential. Our approach is based on the analysis of the first and second moment of the number of overlaps of a given size. We treat both the case of equal sequences (and nonzero shifts) and independent sequences.

math.PR↗

arXiv2002

Thermodynamical Limit for Correlated Gaussian Random Energy Models

Let $\{E_{\s}(N)\}_{\s\inΣ_N}$ be a family of $|Σ_N|=2^N$ centered unit Gaussian random variables defined by the covariance matrix $C_N$ of elements $\displaystyle c_N(\s,τ):=\av{E_{\s}(N)E_τ(N)}$, and $H_N(\s) = - \sqrt{N} E_{\s}(N)$ the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition $N=N_1+N_2$, and all pairs $(\s,\t)\in Σ_N\times Σ_N$: $$ c_N(\s,τ)\leq \frac{N_1}{N} c_{N_1}(π_1(\s),π_1(τ))+ \frac{N_2}{N} c_{N_2}(π_2(\s),π_2(τ)) $$ where $π_k(\s), k=1,2$ are the projections of $\s\inΣ_N$ into $Σ_{N_k}$. The condition is explicitly verified for the Sherrington-Kirckpatrick, the even $p$-spin, the Derrida REM and the Derrida-Gardner GREM models.

math-ph↗