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Let N be the number of triangles in an Erdős–Rényi graph G (n , p) on n vertices with edge density p = d / n , where d > 0 is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean d 3 / 6 as n → ∞ . We address the upper tail problem for N, namely, we investigate how fast k must grow, so that P (N ≥ k) is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when k 1 / 3 log k < (3 2 - o (1)) 2 / 3 log n (sub-critical regime) as well as pin down the tail behavior when k 1 / 3 log k > (3 2 + o (1)) 2 / 3 log n (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately (6 k) 1 / 3 . This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades and culminating in Harel et al. (Duke Math J 171(10):2089–2192, 2022), which analyzed the problem only in the regime p ≫ 1 n. The proofs rely on several novel graph theoretical results which could have other applications. [ABSTRACT FROM AUTHOR]

Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean p$p$. Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph Gn,p$\mathcal {G}_{n,p}$ equipped with independent and identically distributed (i.i.d.) edge‐weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. While certain techniques have been devised for the case when p$p$ is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case p=dn$p = \frac{d}{n}$, that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph Gn,dn$\mathcal {G}_{n,\frac{d}{n}}$without edge‐weights and with Gaussian edge‐weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions whose tail probabilities decay at rate e−tα$e^{-t^\alpha }$ with α>0$\alpha >0$, where the regimes 0<α<2$0<\alpha < 2$ and α>2$\alpha > 2$ correspond to tails heavier and lighter than the Gaussian tail, respectively. While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when α>2$\alpha > 2$. In contrast, in the α<2$\alpha < 2$ case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin–Straus theorem, a classical result from spectral graph theory. As a byproduct of our large deviation results, we also establish the law of large numbers behavior for the largest eigenvalue, which also seems to be new and difficult to obtain using existing methods. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at α=2$\alpha = 2$, that is, corresponding to the Gaussian distribution. [ABSTRACT FROM AUTHOR]

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably by Bhamidi, Bresler, and Sly [Ann. Appl. Probab. 21 (2011), pp. 2146–2170] through the analysis of a natural Heat-bath Glauber dynamics, and by Chatterjee and Diaconis [Ann. Statist. 41 (2013), pp. 2428–2461; Eldan [Geom. Funct. Anal. 28 (2018), pp. 1548–1596; and Eldan and Gross [Ann. Appl. Probab. 28 (2018), pp. 3698–3735] via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase , including a Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W_1-Wasserstein distance to Erdős-Rényi graphs, previously obtained by Reinert and Ross [Ann. Appl. Probab. 29 (2019), pp. 3201–3229], is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime. [ABSTRACT FROM AUTHOR]

We derive the hydrodynamic limit of the Kawasaki dynamics for the one-dimensional conservative system of unbounded real-valued spins with arbitrary strong, quadratic and finite-range interactions. This significantly extends prior results for bounded interaction by Rezakhanlou and complements results obtained by H.T. Yau. The result is obtained by adapting two-scale approach of Grunewald, Otto, Villani and Westdickenberg combined with the authors' recent approach on conservative systems with strong interactions. [ABSTRACT FROM AUTHOR]

We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on R m are replaced by non-commutative laws of m-tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu's non-commutative L 2 -Wasserstein distance using a new type of convex functions. As a consequence, we show that if (X, Y) is a pair of optimally coupled m-tuples of non-commutative random variables in a tracial W ∗ -algebra A , then W ∗ ((1 - t) X + t Y) = W ∗ (X , Y) for all t ∈ (0 , 1) . Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m-tuples is not separable with respect to the Wasserstein distance for m > 1 . [ABSTRACT FROM AUTHOR]

A low-noise and -power readout integrated circuit (IC) for two-electrode electrocardiogram (ECG) recording is developed in this study using a common-mode charge pump (CMCP) for a robust 20-VPP common-mode interference (CMI). Two-electrode ECG recording offers more comfort than three-electrode ECG recording. Contrasting to the three-electrode ECG recording, the two-electrode ECG recording is affected by CMI during measurements; the intervention of a large CMI will distort the ECG signal measurement. To achieve robustness for the CMI, the proposed ECG readout IC adopts CMCP—it uses switched capacitors that store and subtract CMI by control logic. In this paper, a window comparator structure is applied to CMCP to obtain a signal with less distortion. The window voltage ranges were set between the input common-mode ranges in which IA can operate. Therefore, a signal with less distortion was obtained by stopping the operation of CMCP between the window voltage ranges. It also reduced additional current consumption. To achieve this, the proposed circuit is implemented using a chopper stabilization technique. The chopper implemented in the amplifier can reduce low-frequency noise components, such as 1/f noise, and it comprises a CMCP, current feedback instrumentation amplifier, QRS peak detector, relaxation oscillator, voltage reference, timing generator, and serial peripheral interface on a single chip. The proposed circuit was designed using a standard 0.18 μm CMOS process with an active area of 0.54 mm2. The proposed CMCP achieves a CMI robustness of 20 VPP at 60 Hz. The measured input-referred noise level was 119 nV/√Hz at 1 Hz, and the power consumption was 23.83 μW with a 1.8 V power supply. [ABSTRACT FROM AUTHOR]

Large deviation behavior of the largest eigenvalue λ 1 of Wigner matrices including those arising from an Erdős-Rényi random graph G n , p with i.i.d. random conductances on the edges has been the topic of considerable interest. However, despite several recent advances, not much is known when the underlying graph is sparse i.e., p → 0 , except the recent works (Bhattacharya et al., Ann Probab 49(4):1847–1885, 2021and Bhattacharya and Ganguly, SIAM J Discret Math, 2020) which consider the simpler case of the graph without additional edge weights. Under sufficiently general conditions on the conductance distribution, one expects the 'dense' behavior as long as the average degree np is at least logarithmic in n. In this article we focus on the case of constant average degree i.e., p = d n for some fixed d > 0 with standard Gaussian weights. Results in Bandeira and Van Handel (Ann Probab 44(4):2479–2506, 2016) about general non-homogeneous Gaussian matrices imply that in this regime λ 1 scales like log n. We prove the following results towards a precise understanding of the large deviation behavior in this setting. (Upper tail probabilities and structure theorem): For δ > 0 , we pin down the exact exponent ψ (δ) such that P (λ 1 ≥ 2 (1 + δ) log n) = n - ψ (δ) + o (1). Further, we show that conditioned on the upper tail event, with high probability, a unique maximal clique emerges with a very precise δ dependent size (takes either one or two possible values) and the Gaussian weights are uniformly high in absolute value on the edges in the clique. Finally, we also prove an optimal localization result for the leading eigenvector, showing that it allocates most of its mass on the aforementioned clique which is spread uniformly across its vertices. (Lower tail probabilities): The exact stretched exponential behavior P (λ 1 ≤ 2 (1 - δ) log n) = exp - n ℓ (δ) + o (1) is also established. As an immediate corollary, one obtains that λ 1 is typically (1 + o (1)) 2 log n , a result which surprisingly appears to be new. A key ingredient in our proofs is an extremal spectral theory for weighted graphs obtained by an ℓ 1 - reduction of the standard ℓ 2 - variational formulation of the largest eigenvalue via the classical Motzkin-Straus theorem [37], which could be of independent interest. [ABSTRACT FROM AUTHOR]

Featured Application: Low noise circuit, delta-sigma modulator, analog to digital converter. This paper presents an incremental second-order delta-sigma modulator with a coarse-fine input buffer in 180-nm CMOS. The modulator's architecture was implemented as a second-order cascade of integrators with a feedback structure. The switched-capacitor integrator was operated in discrete time, with high-gain amplifiers required to achieve improved performance during the integration phase. The amplifier comprised rail-to-rail input and gain-boosted cascode intermediate stages, thus achieving a high gain and wide input voltage range. The circuit adopts a coarse-fine buffer for higher performance. The coarse buffer is operated first to enable fast settling through a high slew rate, followed by the fine buffer to satisfy the low-noise and high-accuracy characteristics. The fine buffer has a smaller current consumption with higher power efficiency. The experiment results show that the proposed input buffer achieved a 13.14 effective number of bits and an 80.87 dB signal-to-noise and distortion ratio. The modulator operates a single bit and sampling clock at 125 kHz. The proposed delta-sigma modulator was operated at 1.8 V. The proposed circuit was designed using a standard 0.18-μm CMOS process with an active area of 1.06 mm2. The total current consumption with the coarse-fine buffer was 1.374 mA. [ABSTRACT FROM AUTHOR]

This paper presents a low-noise potentiostat readout circuit with a chopper fully differential difference amplifier (FDDA) for glucose monitoring. Glucose monitoring is necessary for the early diagnosis of diabetes complications and for health management. Ammeter electrochemical sensors are widely used for glucose detection, and in general, a three-electrode structure of a reference electrode (RE), a counter electrode (CE), and a working electrode (WE) is implemented with a potentiostat structure. A low-noise characteristic of the readout circuit is essential for highly accurate glucose monitoring. The chopping technique can reduce low-frequency noises such as 1/f noise and can achieve the required low-noise characteristic. The proposed potentiostat readout circuit is based on a low-noise chopper FDDA with a class-AB output stage. The implementation of the chopper FDDA scheme of the potentiostat readout circuit can decrease the number of amplifiers in the control part of the potentiostat, with reduced power consumption and a wide dynamic output range. The negative feedback loop of the inverting amplifier scheme with the FDDA maintains the voltage between the WE and RE constants. The negative feedback loop tracks the reference voltage of the RE with an input voltage of the WE. The proposed potentiostat readout circuit is designed in the standard 0.18 µm CMOS process, and the simulated current consumption is 48.54 μA with a 1.8 V power supply. The simulated input-referred noise level was 8.53 pArms. [ABSTRACT FROM AUTHOR]

In this study, we describe a dual-chopper glitch-reduction current-feedback instrumentation amplifier (CFIA) with a ripple reduction loop. The amplifier employs the chopping technique to reduce low-frequency noise, such as 1/f noise. A glitch caused by chopping occurs at each chopper clock edge and results in intermodulation distortion (IMD). Owing to the input offset, the chopping technique also produces ripples. In this study, the glitch-induced IMD was reduced using a fill-in technique whereby only neat signals were alternately used as outputs by avoiding the glitch section with dual-chopping channel CFIA. To avoid using a high-order, low-frequency filter, a ripple reduction loop was implemented to reduce the ripple generated by chopping. The CFIA is based on a low-noise chopper fully differential difference amplifier with a cascode stage and a Monticelli-class AB output stage, which can drive a larger load and increase power efficiency. The proposed dual-chopper CFIA was fabricated using a 0.18 µm CMOS standard process, and its current consumption with a 1.8-V power supply is 29.5 μA. The proposed CFIA has a gain of 51 V/V, input referred noise of 53.3 nV/√Hz at 1 Hz, and a noise efficiency factor of 4.48. [ABSTRACT FROM AUTHOR]

In this paper, low-noise incremental delta–sigma analog front end (AFE) integrated circuit (IC) for capacitive microsensors is presented. A conventional capacitance-to-digital converter (CDC) mainly uses a multi-stage capacitive sensing amplified stage (CSA) and analog-to-digital converters. The multi-stage CSA is not suitable for application in various Internet of things (IoT) devices that require low power because the power consumption of the analog front-end circuit increases in proportion to the number of amplifiers and the chip area increases. So, the presented delta-sigma AFE can convert the capacitance changes to the digital codes directly. This structure can achieve a small active area and low power consumption. The delta–sigma AFE achieves low-noise and high linearity using a capacitor polarity swapping technique. The measured effective resolution is 16.2 bits, and the non-linearity is 0.05% full-scale output (FSO). The integrated circuit is implemented in a 0.18-µm standard CMOS process. All functional blocks, including the analog circuits (bandgap reference, voltage reference, and delta–sigma capacitance-to-digital converter) and digital block (accumulator and timing generator), are integrated on a chip. The proposed incremental delta–sigma AFE consumes 1.12 mW of power from a 3.3-V supply at a sampling frequency of 500 kHz and occupies a total active area of 0.42 mm2. [ABSTRACT FROM AUTHOR]

This paper presents a potentiostat readout circuit with low-noise and mismatch-tolerant current mirror using chopper stabilization and dynamic element matching (DEM) for electrochemical sensors. Current-mode electrochemical sensors are widely used to detect the blood glucose and viruses in the diagnosis of various diseases such as diabetes, hyperlipidemia, and the H5N1 avian influenza virus (AIV). Low-noise and mismatch-tolerant characteristics are essential for sensing applications that require high reliability and high sensitivity. To achieve these characteristics, a proposed potentiostat readout circuit is implemented using the chopper stabilization scheme and the DEM technique. The proposed potentiostat readout circuit consists of a chopper-stabilized programmable gain transimpedance amplifier (TIA), gain-boosted cascode current mirror, and a control amplifier (CA). The chopper scheme, which is implemented in the TIA and CA, can reduce low frequency noise components, such as 1/f noise, and can obtain low-noise levels. The mismatch offsets of the cascode current mirror can be reduced by the DEM operation. The proposed current-mirror-based potentiostat readout circuit is designed using a standard 0.18 μm CMOS process and can measure the sensor current from 350 nA to 2.8 μA. The input-referred noise integrated from 0.1 Hz to 1 kHz is 21.7 pARMS, and the power consumption was 287.9 μW with a 1.8 V power supply. [ABSTRACT FROM AUTHOR]

In this paper, a low-power and low-noise readout circuit for resistive-bridge microsensors is presented. The chopper-stabilized, recycling folded cascode current-feedback instrumentation amplifier (IA) is proposed to achieve the low-power, low-noise, and high-input impedance. The chopper-stabilized, recycling folded cascode topology (with a Monticelli-style, class-AB output stage) can enhance the overall noise characteristic, gain, and slew rate. The readout circuit consists of a chopper-stabilized, recycling folded cascode IA, low-pass filter (LPF), ADC driving buffer, and 12-bit successive-approximation-register (SAR) analog-to-digital converter (ADC). The prototype readout circuit is implemented in a standard 0.18 µm CMOS process, with an active area of 12.5 mm2. The measured input-referred noise at 1 Hz is 86.6 nV/√Hz and the noise efficiency factor (NEF) is 4.94, respectively. The total current consumption is 2.23 μA, with a 1.8 V power supply. [ABSTRACT FROM AUTHOR]

Air flow measurements provide significant information required for understanding the characteristics of insect movement. This study proposes a four-channel low-noise readout integrated circuit (IC) in order to measure air flow (air velocity), which can be beneficial to insect biomimetic robot systems that have been studied recently. Instrumentation amplifiers (IAs) with low-noise characteristics in readout ICs are essential because the air flow of an insect's movement, which is electrically converted using a microelectromechanical systems (MEMS) sensor, generally produces a small signal. The fundamental architecture employed in the readout IC is a three op amp IA, and it accomplishes low-noise characteristics by chopping. Moreover, the readout IC has a four-channel input structure and implements an automatic offset calibration loop (AOCL) for input offset correction. The AOCL based on the binary search logic adjusts the output offset by controlling the input voltage bias generated by the R-2R digital-to-analog converter (DAC). The electrically converted air flow signal is amplified using a three op amp IA, which is passed through a low-pass filter (LPF) for ripple rejection that is generated by chopping, and converted to a digital code by a 12-bit successive approximation register (SAR) analog-to-digital converter (ADC). Furthermore, the readout IC contains a low-dropout (LDO) regulator that enables the supply voltage to drive digital circuits, and a serial peripheral interface (SPI) for digital communication. The readout IC is designed with a 0.18 μm CMOS process and the current consumption is 1.886 mA at 3.3 V supply voltage. The IC has an active area of 6.78 mm2 and input-referred noise (IRN) characteristics of 95.4 nV/√Hz at 1 Hz. [ABSTRACT FROM AUTHOR]