We consider the spherical spin glass model defined by a combination of the pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian and the ferromagnetic Curie-Weiss Hamiltonian. In the large system limit, there is a two-dimensional phase diagram with respect to the temperature and the coupling strength. The phase diagram is divided into three regimes; ferromagnetic, paramagnetic, and spin glass regimes. The fluctuations of the free energy are known in each regime. In this paper, we study the transition between the ferromagnetic regime and the paramagnetic regime in a critical scale. 1 arXiv:1805.05630v1 [math.PR] 15 May 2018 E T J 1 2β 0 1 1 spin glass paramagnetic ferromagnetic Figure 1: Phase diagram for SSK+CW model. Here, β is the inverse temperature and J is the coupling constant.regime) or J (ferromagnetic regime). The limiting free energy is analytic with respect to both β and J in each regime, but not on the boundary.Recently, the authors of [6] showed that the result of Kosterlitz, Thouless, and Jones is rigorous. Furthermore, the authors also evaluated the distribution of the fluctuations of the free energy in each regime. (The case when J = 0 was obtained earlier in [4].) The order of the fluctuations are N −2/3 , N −1 , N −1/2 and the limiting distributions are Tracy-Widom, Gaussian, and Gaussian in the spin glass, paramagnetic regime, ferromagnetic regime, respectively. In the same paper, the transition between the spin glass regime and the ferromagnetic regime was also studied. However, the other two transitions and the triple point were left open. The goal of this paper is to describe the transition between the paramagnetic regime and and the ferromagnetic regime.Another system which combines a spin glass and a ferromagnetic model is the SSK with an external field. The difference between the CW Hamiltonian and an external field is that one is a quadratic function and the other is a linear function of the spin variables. These two models are related; see [12] for a one-sided inequality. For the spin glass with external field, the fluctuations of the free energy were computed recently in [13,14] when the coupling constant is positive (for both SSK and SK (Sherrington-Kirkpatrick) cases with general spin interactions). However, the transitions are not obtained except for certain large deviation results [18,16]. One of the interests of the SSK+CW model is that it is an easier model which can be analyzed in detail in the transitional regimes.(1.11)1 In [6], we consider the case when the diagonal entries of M have mean J N and the off-diagonal entries have mean J N where J and J are allowed to be different. However, in this case, M = 1 √ N + J N 11 T + J −J N I where I is the identity matrix. This only shifts all eigenvalues by a deterministic small number. As we will see in Remark 2.2, it is not more general than the case with J = J.

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ImportanceCasting is recommended for adults older than 65 years with distal radius fractures (DRFs) because similar long-term outcomes are achieved regardless of treatment. However, physiologically younger adults could benefit from operative DRF management despite advanced chronologic age.ObjectiveTo examine how chronologic age compares with measures of physiologic age in DRF treatment recovery.Design, Setting, and ParticipantsThis retrospective secondary analysis of the Wrist and Radius Injury Surgical Trial (WRIST) was performed from May 1 to August 31, 2022. WRIST was a 24-center randomized clinical trial that enrolled participants older than 60 years with unstable DRFs from April 1, 2012, to December 31, 2016.InterventionsParticipants selected casting or surgery. Patients who selected surgery were randomly assigned to volar lock plating, percutaneous pinning, or external fixation. Participants were stratified by chronologic age, number of comorbidities, and activity status.Main Outcomes and MeasuresThe primary outcome was Michigan Hand Outcomes Questionnaire (MHQ) score assessed at 6 weeks, 3 months, 6 months, and 1 year. Partial correlation (PC) analysis adjusted for confounding.ResultsThe final cohort consisted of 293 participants (mean [SD] age, 71.1 [8.89] years; 255 [87%] female; 247 [85%] White), with 109 receiving casting and 184 receiving surgery. Increased chronologic age was associated with increased MHQ scores in the surgery group at all time points but decreased MHQ scores in the casting group at 12 months (mean [SD] score, −0.46 [0.21]; P = .03). High activity was associated with improved MHQ scores in the surgical cohort at 6 weeks (mean [SD] score, 12.21 [5.18]; PC = 0.27; P = .02) and 12 months (mean [SD] score, 13.25 [5.77]; PC = 0.17; P = .02). Comorbidities were associated with decreased MHQ scores at all time points in the casting group. Clinically significant differences in MHQ scores were associated with low physical activity, 4 or more comorbidities, or increased age by 15 years.Conclusions and RelevanceIn this retrospective secondary analysis of WRIST, chronologic age was not associated with functional demand. These findings suggest that physicians should counsel active older adults with few comorbidities on earlier return to daily activities after surgery compared with casting.Trial RegistrationClinicalTrials.gov Identifier: NCT01589692

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