ترغب بنشر مسار تعليمي؟ اضغط هنا

Ising Model Observables and Non-Backtracking Walks

44   0   0.0 ( 0 )
 نشر من قبل Tyler Helmuth
 تاريخ النشر 2012
  مجال البحث
والبحث باللغة English
 تأليف Tyler Helmuth




اسأل ChatGPT حول البحث

This paper presents an alternative proof of the connection between the partition function of the Ising model on a finite graph $G$ and the set of non-backtracking walks on $G$. The techniques used also give formulas for spin-spin correlation functions in terms of non-backtracking walks. The main tools used are Viennots theory of heaps of pieces and turning numbers on surfaces.

قيم البحث

اقرأ أيضاً

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $ G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $log_k n$. The high dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of $G$. Via these, we show that operators $T$ as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property ($r$-normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group $G$, previously known for groups of type $widetilde A_n$ and $widetilde C_2$.
In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.
We study a two dimensional Ising model between thermostats at different temperatures. By applying the recently introduced KQ dynamics, we show that the system reaches a steady state with coexisting phases transversal to the heat flow. The relevance o f such complex states on thermodynamic or geometrical observables is investigated. In particular, we study energy, magnetization and metric properties of interfaces and clusters which, in principle, are sensitive to local features of configurations. With respect to equilibrium states, the presence of the heat flow amplifies the fluctuations of both thermodynamic and geometrical observables in a domain around the critical energy. The dependence of this phenomenon on various parameters (size, thermal gradient, interaction) is discussed also with reference to other possible diffusive models.
We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$t h Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.
Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exis ts. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $Delta$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $Delta$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $Theta(log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا