Communications in Mathematical Physics最新文献

Fluctuation problems in mean-field spin glass theory enjoy a rich history. In this paper, we investigate this problem in the high-temperature regime by analyzing the fluctuation of the partition function in the general mixed p-spin glass models under the weak external field assumption: . By extending the cluster expansion approach to this generic setting, we convert the fluctuation problem as a hypergraph counting problem and thus obtain a new multiple-transition phenomenon in terms of the fluctuation of the partition function. All our fluctuation results hold up to an explicit second moment threshold (beta _f), which is away from the static phase transition point (beta _c). Combining this with multivariate Stein’s method for exchangeable pairs, we also obtain an explicit convergence rate under appropriate moment assumptions on the general symmetric disorder coupling. Our results have several further implications. First, our approach works for both even and odd pure p-spin models. The leading cluster structures in the odd p case are different and more involved than in the even p case. This gives a combinatorial explanation for the folklore that odd p-spin is usually more complicated than even p. Second, in the mixed p-spin setting, the cluster structures differ depending on the relation between the minimum effective even and odd p-spins: (p_e) and (p_o), respectively. As an example, at (h=0), there are three sub-regimes: (p_e<p_o, p_o<p_e<2p_o, p_egeqslant 2p_o), wherein the first and third ones, the mixed p-spin model behaves essentially like a pure p-spin model, and only in the second regime, it is more like a mixture. This presents another criterion for classifying mean-field spin glass models compared to the work of Auffinger and Ben Arous (Ann. Probab. 41 (2013), no. 6, 4214–4247), where the idea is based on complexity computations for spherical models. Third, our framework naturally implies a multi-scale fluctuation phenomenon conjectured in the work of Bovier and Schertzer (Probab. Theory Related Fields 189 (2024), no. 3–4, 771–810), in the (h=0) case for pure p-spin models. Our results suggest that this phenomenon holds for general mixed p-spin models under weak external fields, including (h=0). We also discuss the extension to general multi-species mixed p-spin models.

We develop an Ornstein–Zernike theory for the two-dimensional random-cluster model with (1 le q <4) that also applies in its near-critical regime. In particular, we prove an asymptotic formula for the two-point function which holds uniformly for (p < p_c) and blends the subcritical and near-critical behaviours of the model. The analysis is carried out by studying the renewal properties of a subcritical percolation cluster, at the scale of the correlation length. More precisely, we explore sequentially the cluster in a given direction, by slices of thickness comparable to the correlation length. We show that this exploration satisfies the properties of a killed Markov renewal process—a class of processes that may be analysed independently and have Brownian behaviour. In addition to the two-point function estimate, we derive other consequences of the Ornstein–Zernike theory such as an invariance principle for the rescaled cluster and the strict convexity of the inverse correlation length—all at the scale of the correlation length, uniformly in (p<p_c). Finally, our approach differs from that of earlier papers of Campanino, Ioffe, Velenik and others, with the cluster being dynamically explored rather than constructed from its diamond decomposition.

The symmetric inclusion process (SIP) models particles diffusing on a graph with mutual attraction. We recently showed [KS24] that, in the log-concave regime (where diffusivity dominates interaction), the spectral gap of the conservative SIP matches that of a single particle. In this paper, our main result demonstrates that this identity generally fails outside this regime, but always holds for the non-conservative SIP, regardless of the interaction strength. When this one-particle spectral gap identity breaks down, we derive sharp bounds for the gap in terms of diffusivity, and reveal a two-particle spectral gap identity in the vanishing diffusivity limit. Our approach leverages the rigid eigenstructure of SIP, refined comparisons of Dirichlet forms for arbitrary diffusivity and particle numbers, and techniques from slow–fast system analysis. These findings extend to the dual interacting diffusion known as Brownian energy process, and shed some light on the spectral gap behavior for related Dirichlet-reversible systems on general, non-mean-field, geometries.

For stochastic Hamilton–Jacobi (SHJ) equations, instability points are the space-time locations where two eternal solutions with the same asymptotic velocity differ. Another crucial structure in such equations is shocks, which are the space-time locations where the velocity field is discontinuous. In this work, we provide a detailed analysis of the structure and relationships between shocks, instability, and competition interfaces in the Brownian last-passage percolation model, which serves as a prototype of a semi-discrete inviscid stochastic HJ equation in one space dimension. Among our findings, we show that the shock trees of the two unstable eternal solutions differ within the instability region and align outside of it. Furthermore, we demonstrate that one can reconstruct a skeleton of the instability region from these two shock trees.

When the local dimension d is an odd prime, the qudit Clifford group is only a 2-design, but not a 3-design, unlike the qubit counterpart. This distinction and its extension to Clifford orbits have profound implications for many applications in quantum information processing. In this work we systematically delve into general qudit Clifford orbits with a focus on the third moments and potential applications in shadow estimation. First, inspired by shadow estimation, we extend the concept of shadow norm from observables to a state ensemble and use this figure of merit to gauge the performance of the ensemble in shadow estimation in the worst-case scenario. Then, we show that the third normalized frame potential and shadow norm are both (mathcal {O}(d)) for any Clifford orbit, including the orbit of stabilizer states, although the operator norm of the third normalized moment operator may increase exponentially with the number n of qudits when (dne 2mod 3). Moreover, we prove that the shadow norm of any magic orbit is upper bounded by the constant 15/2, and a single magic gate can already eliminate the (mathcal {O}(d)) overhead in qudit shadow estimation and bridge the gap between qudit systems and qubit systems. Furthermore, we propose simple recipes for constructing approximate and exact 3-designs (with respect to three figures of merit simultaneously) from one or a few Clifford orbits. Notably, accurate approximate 3-designs can be constructed from only two Clifford orbits. For an infinite family of local dimensions, exact 3-designs can be constructed from two or four Clifford orbits.

Inspired by parallel developments in coarse geometry in mathematics and exact macroscopic quantization in physics, we present a family of general trace formulas which are universally quantized and depend only on large-scale geometric features of the input data. They generalize, to arbitrary dimensions, formulas found by Roe in his partitioned manifold index theorem, as well as the Kubo and Kitaev formulas for 2D Hall conductance used in physics.

Let (A=(A_v)_{vin mathcal {T}}) be the balanced Gaussian Branching Random Walk on a d-ary tree (mathcal {T}) and let (M^A) be the multiplicative chaos with parameter (gamma in (0, sqrt{2log d})) constructed from A. In this work we establish the precise first order asymptotics of negative exponential moment of (M^A), i.e. we prove that for (t_k = lambda {p}^k) with (lambda >0) and ({p}) an explicit constant depending only on (gamma ) and d, we have as (k rightarrow infty ),

where (h:(0,infty )rightarrow mathbb {R}) is a non-explicit positive continuous function. This result allows us to study the law of A tilted by (e^{-t_k M^A}) for particular values of (lambda ), with (krightarrow infty ). In this setting we prove that the normalized (L^1) norm of A in generation (k-a) is bounded and converges to 0 when first (krightarrow infty ) and then (arightarrow 0). As an application we prove that in this setting, under the tilt (e^{-t_k M^A}) and with (krightarrow infty ), the Branching Random Walk A exhibits a weak decay of correlations, which is not present in the non-tilted model. Our methods also apply to the usual Branching Random Walk ((S_v)_{vin mathcal {T}}) and with (M^A) replaced by (frac{1}{2}(M^{+}+ M^{-})), where (M^{+}) and (M^{-}) are the multiplicative chaoses with parameter (gamma in (0, sqrt{2log d})) constructed from S and (-S). In that case we prove that, as (krightarrow infty ),

where (tilde{h}:(0,infty )rightarrow mathbb {R}) is again a non-explicit positive continuous function. Our models are motivated by Euclidean field theory and can be seen as hierarchical versions of the massless Liouville and the sinh-Gordon field theory in infinite volume. From this perspective our analysis sheds new light on the existence and the decay or correlations in these models, which are among the major open questions in this area.

We study Lie conformal algebroids (LCAd) and their representations using the language of (lambda )-brackets and Lie conformal algebras. We describe several general constructions, such as the LCAd of conformal derivations ({{,textrm{CDer},}}(A)) of a differential algebra A, the gauge LCAd (mathcal {G}(M,E)) associated to a differential algebra A, its module M, and an LCAd E, the current LCAd (widehat{F}) of a Lie algebroid F, and the LCAd structure of the space (Omega (mathcal {V})) of Kähler differentials over a Poisson vertex algebra (PVA) (mathcal {V}). We develop the cohomology theories of LCAd and we relate them to the corresponding cohomology theories of PVA. In particular, we find an isomorphism between the cohomology of a PVA (mathcal {V}) with coefficients in a module M and the corresponding cohomology of the LCAd (Omega (mathcal {V})) with coefficients in the same module.

We use a 2-categorical version of (de-)equivariantization to classify (3+1)d topological orders with a finite G-symmetry. In particular, we argue that (3+1)d fermionic topological orders with G-symmetry correspond to nondegenerate (textbf{2SVect})-central G-crossed braided fusion 2-categories. We then show that the categorical data necessary to define these theories agrees with that arising from a fermionic generalization of the Wang–Wen–Witten construction of bosonic topological theories with G-symmetry saturating an anomaly.

We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength (beta >0) couples the particles and the substrate. Our first main result proves crystallization for all (beta >0), corresponding to the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations. Our second main result concerns fluctuations estimates for (beta in (0,1)). We obtain bounds on the distance between distinct minimizers with the same number N of particles, showing a sharp scaling law (N^{3/4}) when (beta ) is rational, and (N^{1/3}) when (beta ) is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit.